Chapter 4 The Representation Theory of Several Special Groups

Cbapter 4

The Representution The09 of SeveruZ Specid Grozips

This chapter is devoted to the representation theory of the symmetric group S,, of all permutations on a set of n objects and to that of the complex general linear group GL(V) of all nonsingular linear transformations on an rn-dimensional complex space V as well as certain of its subgroups. A complete treatment is a hard task, well beyond the scope of this introductory book. Consequently, our discussion is sometimes incomplete and without proofs. However, there is a general interest in the representations of these groups among people outside of mathematics for many of whom the standard treatments are either too protracted or too difficult. This last chapter is an attempt to present a readable account of various topics for such an audience. Even among students of mathematics there is doubtless a sizable group who would like to know something of the general situation before attempting the arduous program required for a rigorous and detailed understanding. The first section of the chapter is devoted LO the development of the ordinary representation theory of S,,. The presentation is based on an approach discovered by A. Young. His principal tools are the Young’s tableau and the Young’s frame. The ideas of these constructs are introduced and applied to deducing the irreducible representations of S,, . Since the methods prove to be lengthy, certain easier methods are considered for special cases. The second section is concerned with symmetric algebras and modules over symmetric algebras. The discussion of these kinds of algebras is largely 213

214

4. Represetitation Theory of Special Groups

limited to the case of the group algebra of a finite group. They are introduced to provide a framework in which to discuss the relation between the representations of S, and certain special representations of GL(V). Section 3 i s an application of the results of Section 2 to the calculation of the integral representations of GL(V). This connection is established by introducing the group S,, as left operators on the n-fold tensor product . 0V and the group GL(V) as right operators. There proves to exist a V 0.. natural duality between right GL(V)-submodules of the tensor product and right ideals of the group algebra A of S,. In the last section we treat various details about the representation theory of what are called the standard matrix groups. They are the complex general linear group GL,,(K) of all nonsingular, complex rn x i x matrices and certain of its subgroups. An effort is made to use these groups as a n introduction to the subject of Lie groups without benefit of the standard definitions. A bare minimum of topology and differential geometry is introduced. The concept of Lie algebra is developed by means of the example of the three-dimensional rotation group SO(3). A sketch is made of the classification of semisimple Lie algebras by means of their roots. Then irreducible modules and their weight diagrams are discussed. Examples are supplied in the cases of SO(3), SU(2), a n d SU(m). 1. THE REPRESENTATION THEORY OF THE SYMMETRIC GROUP

The most recent detailed treatment of the representation theory of the symmetric group i s that of Robinson (1961) to whom we must refer the reader interested in more than a n outline. Our goal is merely to present the high 1 ights. The ordinary representation theory of the symmetric groups was first worked out by Frobenius, but our presentation follows a n approach discovered by Alfred Young independently of Frobenius. The method of Young is based on a detailed analysis of the group algebra A of the symmetric group S,. Young invented a procedure, not using the theory of characters, for determining the primitive idempotents of A. The fundamental ideas hinge on the concept of a Young’s tableau which is described below. The section contains a discussion of the relations between partitions of an integer n , the Young’s frames and Young’s tableaux belonging to n, and the irreducible representations of the symmetric group S,. A method is explained of determining the primitive idempotents of the simple components of the group algebra A = KS,, of the symmetric group S,, over the complex numbers. A useful method of computing the value of the character of a n irreducible representation associated with a frame is given. A procedure for finding the matrix of a transposition ( r , r 1) in the Young’s semirational

+

1. The Representation Theory of the Symmetric Group

21s

irreducible representation is described. Finally, a sketch of the method of computation of the matrices of Young’s integral representation is included.

(1.1) DEFINITION. A partition of the positive integer n is a sequence {mi}, 1I i i k , of positive integers such that m i + l 2 mi, 1 2 i k - 1, and m , + . . . + mk = n. The sequence (3, 2, l} is a partition of 6. It is customary to order the partitions of n such that {mi} > {m,’}if, for the first j such that mi # mj‘, one has mj > mj‘. The ordered partitions of 5 are (5) > (4, l} > {3,2) > (3, 1, 1) > (252, 11 > ( 2 , 1, 1, 1) > (1, 1, 1, 1, 1). (1.2) DEFINITION. A frame F corresponding to the partition {m,}, 1 < i I k , of the positive integer n is a diagram consisting of k left-justified rows of empty square boxes, the ith row of which contains m iboxes. Such a frame is said to belong to the integer n. For example, the frame F corresponding to the partition (3, 2, I} of 6 is as shown in Fig. (1.3). There is no known

function J’ such that f (n) is the number of frames belonging to the integer n. However, tables are available which list this number for fairly extensive ranges of n. Let F and F‘ belong to the partitions {mi} and {m,’},respectively, of n. Then F > F‘ if and only if {mi} > {mi’}.

(1.4) DEFINITION. A tableau T corresponding to the frame F belonging to n is a diagram in which the distinct integers from 1 through n have been inserted into the frame F. Such a tableau is said to belong to F and also to the integer n. The canonical tableau T belonging to the frame F of Fig. (1.3) is as shown. There are 719 = 6! - 1 other tableaux associated with this frame F. These are obtained by performing all possible permutations on the entries of the tableau T of Fig. (1.5). They are mutually congruent in the sense that

given any two of them, say T’and T , there exists an s E S , such that s T = T . The precise action of s on T‘ is defined below. The notion of a canonical tableau for any frame should be clear to the reader. Note that the canonical tableau serves to label the squares of the associated frame F.

216

4. Representation Theory of Special Groups

It is convenient to simplify the notation for frames and tableaux. Either the symbol

*** ** *

or the symbol

... ..

can be used t o denote the frame F of Fig. (1.3) and a similar scheme used for any frame. The marks * or . replacing the squares are called nodes. A symbol such as 456 31 2

is used to denote a tableau belonging to the frame F. An element r of the symmetric group S, acts on any tableau T belonging to n. To illustrate this, let s = (123)(45) be an element of S, and let T denote the tableau 645 31 . 2

Then the tableau sT generated by the action of s on T is 654 12 . 3

The action of r E S , on the canonical tableau is defined by Fig. (1.6).

The effect of r E S,, on a tableau T belonging to n can be thought of as changing the names of the squares of T. Square i of T becomes square r(i) of rT. According to this viewpoint, r effects an alias transformation on T. However, one can also consider the geometric result of applying r to T whose entries are considered to move under the action of r. For example, the permutation s = (123)(45) transforms the tableau

123 45 6

into the tableau

23 1 54 6

1. The Representation Theory of the Symmetric Group

217

Observe that the entry 1 moves from its first position to the third, the entry 2 from its second position to the first, etc. One notes that 1 moves to the square occupied by s - l ( l ) , 2 moves to the square occupied by s-'(2), and, in general, the entry i to the square occupied by s-'(i). Thus if r E S,, is an alibi transformation moving the elements of the tableau T belonging to n, then the element i of Tmoves to the square of Tcontaining r-'(i). Given any tableau T with n elements, one can make the following definition.

(1.7) DEFINITION. The row-group P ( T ) of Tis the set of allp E S,, such that p does not transform any element i of Tout of its row. The column-group Q(T) of T is the set of q E S,, ,no member of which moves any element of T out of its column. It should be clear that P ( T ) and Q(T) are subgroups of S,, . The row-group P ( T ) of the tableau T of Fig. (1.5) is generated by the set {(12), (13), (23), (45)) of transpositions and the column group Q ( T )by the set W4),

(46), (25)).

The subgroups P ( T ) and Q(T) play an important role in the analysis of the representations of S,, . There is associated with P ( T ) the element P = p , p E P ( T ) , of the group algebra A = KS,, of the symmetric group S,, and with Q(T) the element Q &(q)q,q E Q(T), where c(q) is 1 for even and - 1 for odd q. Each tableau T belonging to n determines a unique element e(T) = PQ of the group algebra A according to the above definitions. Let p , p' E P ( T ) and q, q' E Q(T) with pq = p'q'. Then one has that p ' - l p = 4'q-l belongs to P ( T ) n Q ( T ) which implies that it is 1. Thus p = p ' and q = q' so that the element e(T) can be written in the form

=c

1

(1.9) REMARK.For each choice of the tableau T belonging to the frame F, the element e(T) is essentially idempgtent, that is, it differs from an idempotent in A by a scalar multiplication. As a, matter of fact, e(T) is essentially a primitive idempotent so that the left ideal Ae(T) is a minimal left ideal in the group algebra A. Furthermore, Ae(T) and Ae(T') are A-isomorphic minimal left ideals of A if and only if T and T' belong to the same frame F. It is a wellknown fact, which we discuss in slightly more detail below, that the number of classes of conjugate elements of S,, is equal to the number of distinct partitions of n, that is, to the number of distinct frames F which belong to n. Consequently, the set of minimal left ideals {Ae(T)), one T to each frame, is a full set of representative elements from the classes of isomorphic irreducible S,,-modules of the symmetric group S,, over the complex numbers. The proof of these facts is highly computational and intimately connected with the properties of the tableaux and the action of the symmetric group on them.

218

4. Representation Theory of Special Groups

Our treatment relies strongly on the works of Boerner (1963) and of Curtis and Reiner (1962) where various details are treated in greater depth.

I t is necessary to establish a relation between the row and column groups of a tableau T and those of a tableau T' = s T congruent to it. Consider an example to illustrate the desired definition. (1.10) EXAMPLE. The two tableaux T and T', given by

132 54 6

312 64 , 5

and

are congruent under the permutation s = (13)(56) of S, . The permutation r = (123)(45) effects an alibi transformation on the first of them such that

132 213 54 4 4 5 . 6 6 On the other hand, the permutation r' transformation on T ' such that

= srs-'

= (132)(46)

effects an alibi

312 231 64 +46 . 5 5 Note that the element in the first position of either T o r T' goes into the second position: the element in the second position goes into the third, etc. T h u s the permutation r' transforms the tableau T' in a manner parallel to that by which the permutation r transforms T.

(1.1 1) DEFINITION. The permutation r' E S,, is said to be congruent with respect to s to the permutation r E S,, if and only if r' effects the same alibi transformation on T' = sT that r effects on T. = s T b e conkruent to Tunder the permutation s. If r is an alibi transformation from T to rT, then the transformation r' = srs-l is an alibi transformation from T' to r ' T ' congruent to r.

(I.12) LEMMA.Let T'

Proof: Note that any element a E T moves to the square occupied by r - ' ( u ) under the alibi transformation r. The element a' of T' occupying the same square as u is s(a), while the element of T' occupying the same square as

r - ' ( a ) is s r - ' ( a ) . The action of r' on a' is to move it to the square occupied by r ' - ' ( u ' ) = sr-lLY-'(sa)= s r - ' ( u ) which establishes the result. This Lemma has a useful corollary.

1. The Representation Theory of the Symmetric Group

219

(1.13) COROLLARY. For any tableau T belonging to n and any s E S,,, we have P(sT) = sP(T)s-’, Q(sT) = sQ(T)s-’, and e(sT) = se(T)s-’. Proof. The group P ( T ) consists of all elements of S,, which preserve the rows of T under alibi transformations. By Lemma (1.12), sP(T)s-’ must be a subgroup of S,, which preserves the rows of T’ = sT. Thus sP(T)s-’ c P(sT). Starting with sT, one finds that s-’P(sT)s c P ( T ) , which gives the inclusion P(sT) c sP(T)s-’, so that P(sT) = sP(T)s-’. The arguments for Q(sT) and e(sT) are similar. (1.14) LEMMA. Let T be a tableau belonging to n. An element s E S,, is of the form s = pq, p E P(T), q E Q(T), if no two collinear symbols of T are cocolumnar in sT.

Proof. Suppose that s = pq, p E P(T), q E Q(T),and T‘ = pqT = (pqp-‘)pT. If a, b are in r o w j of T, they are in r o w j ofpTsince p is a row transformation on T. Since pqp-’ is a column transformation of p T by Lemma (1.12), it follows that a, b are in different columns of ( p q p - ’ ) p T = p q T . Now suppose that no collinear pair a, b of Tis cocolumnar in sT. Then no two elements of the first column of sT occur in the same row of T. Consequently, there exists p 1 E P ( T ) such that p,Ta nd sT have the same elements in their first columns. Furthermore, no pair a, b collinear in p1Tis cocolumnar in sT. Thus the elements of the second column of sT occur in different rows of p , Tand not in the first column ofp, 7’. Sincep, Tand Thave the same elements in each row, there exists a p 2 E P ( T ) such that p z does not move the first column of p , T and such that p 2 p ,T and sT have the same elements in each of their first two columns. After a finite number of repetitions of this argument, one finds a p E P ( T ) such that each column of p T and sT contain the same elements. Hence there exists a q’ E Q(pT)such that sT = q‘pT. Since Q(pT) = pQ(T)p-’, one has q’ = p q p - l , q E Q(T), and s T = q’pT= p q T , so that s = pq, P E W), E Q(T).

(1.15) LEMMA. Let T and T‘ be two tableaux associated with the partitions { m l , . . . , m,} and {m,’, ..., mt’}, respectively, of n, where the partition { m l ,..., m,) is greater than the partition {m,’, ..., m,’}. Then one has e(T’)e(T)= 0. Proof. The basic claim is that there are two elements a, b which are collinear in T and co-columnar in T’. If this be false, then m,’ 2 m l , otherwise some pair a, b of integers that occur in the first row of Twould occur in the same column of T’. Consequently, Q(T‘) contains an element q’ such that each column of q‘T’ contains the same elements as the corresponding column of T’ while the first row of q’T’ contains the same integers as the first row of T. Furthermore, no collinear pair of Tis co-columnar in q‘T‘. One now considers the second row of T and compares it with the second row of q’T’. This gives

220

4. Representation Theory of Special Groups

m2‘ 2 m z . One repeats the argument to discover that m i = mi‘,1 5 i I r, contradicting the assumption that { m l , . . . , m,} is greater than { m l ’ , . . . , m,’}. Thus there exists a pair a, b collinear in T and co-columnar in T’. The transposition t = (ab) belongs both to P ( T ) and to Q(T’).Thus one has e(T’)t = -e(T’) and t e ( T ) = e ( T ) from which it follows that e( T’)e( T ) = e( T’)tte( T ) = - e(T’)e(T ) .

Consequently, e(T’)e(T)= 0, as was to be shown.

(1.16) LEMMA. Let x be an element of the group algebra A of the symmetric group S,. Suppose there exists a tableau T, belonging to n, such that pxq = &(q)xfor all p E P ( T ) , q E Q(T). Then there is a complex number CI such that x = .*e(T). Proof. Let T be such a tableau for the element x = one has, for p E P(T), q E Q(T),

c &(q)x(s)s

= E(q)x =

c x(r)r, r

E

S,, . Then

c x(r)p-’rq-’ c x(psq)s. =

Thus one obtains

x(psq) = E ( ~ ) X ( S ) ,

(1.17)

For s

=

P EPV), 4 E Q W -

1 , this gives

(1.18)

x(pq) = & ( 4 ) ~ ( 1 ) , P E P ( T ) , 4

E

QV).

Equation (1.18) gives the desired result providing x(s) = 0 when s is not of the form pq. When s is not a pq, Lemma (1.14) implies there exists a pair a, b collinear in T and co-columnar in sT. The transposition t = (ab) is an element both of P ( T ) and also of Q(sT) = sQ(T)s-l. Thus there exists a transposition q-’ in Q(T) such that t = sq-ls-l which implies that s = tsq,

t E P(T),

q E Q(T).

It follows from (1.17) that

x(s) = x(tsq) = &(q)X(s)= -x(s). Thus x ( s ) = 0 unless s is a pq. This gives x

=

c

=

c x(Pq)Pq c &(q)x(l)Pq

as was to be shown.

=

=X U )

c &(4)P4

= x(l)e(T),

(1.19) LEMMA.Let T be a tableau belonging to n. The element e ( T ) is essentially idempotent in the group algebra A of the symmetric group S,, .

Proof. Note that (e(T))’ = PQPQ so that p(e(T))2q= pPQPQq = &(q)(c(T))’for p E P ( T ) , q E Q(T). It follows from Lemma (1.16), that

1. The Representation Theory of the Symmetric Group

221

(e(T))' = Ae(T). To show that 3, is not 0, consider the linear transformation L on the group algebra A = KS, defined by right-translation by e(T). Since &(q)pq,it follows that the matrix of L with respect to the e(T) = 1 natural basis {I = gl, . . . , go), a = n ! , has all 1's along the principal diagonal. Thus one has tr(L) = n!. On the other hand, let the vectors vi = a,e(T), 1 5 i _< p, spanning the nonzero left ideal Ae(T), be the first p elements of a basis B = {vi}, 1 5 i I n ! , of A. Then the matrix of L with respect to the basis B has the form

+ c,,+l

p+ 1 column

since Lvi = ai(eT)(eT) = Aai(eT) = Avi, 1 5 i I p, and L maps A onto Ae(T). Thus one has t r ( t ) = E$', where p is the complex dimension of Ae(T). Consequently, l p = n ! so that E. and p are each positive integers dividing the order of S, . The element e(T)//? is an idempotent in the group algebra A of S,, . (1.20) LEMMA.The left ideal Ae(T) is a minimal left ideal in the group algebra A of the symmetric group S, for any tableau T belonging to n . Proof. If Ae(T) is not minimal, it is the direct sum M @ N of proper left ideals of A. There exists a decomposition of the idempotent e = e(T)/l given by e=f+g, where f and g are nonzero, orthogonal idempotents with f = efe, g = ege. Thus pfq = &(q)f,pgq = &(q)g for p E P ( T ) , q E Q(T). It follows that f and g are nonzero multiples of e(T). This implies in turn that fg is nonzero, a contradiction. Hence Ae(T) is a minimal left ideal of A. This completes the argument. Let T and T' belong to the frame F with T' = sT for s E S, . Then e(T') = se(T)s-', so that e(T')s = se(T) # 0. Consequently, right-translation by s is a nontrivial A-homomorphism of Ae(T') into Ae(T). Since each of these left ideals is minimal, they are isomorphic. On the other hand, let T and T' belong to the frames F and F', respectively, with F > F'. Any isomorphism h of Ae(T') onto Ae(T) is a right-translation according to Theorem (3.13), Chapter 2. In particular, e(T) = ae(T')x b(t)t, where b = 1 b(t)t generates h.

222

4. Representation Theory of Special Groups

Hence,

e ( T ) = ae(T’)x b(t)t = ae(T’)x b(t)te(T)/A = b(t)at[ t - ‘e(T’)t]e(T)/E.. However. [t-le(T’)t]e(T) = e(T”)e(T) = 0, t E S,, , TI‘ = t-’T’, according to Lemma (1.15). This implies that e ( T ) = 0, which is a contradiction. Thus e ( T ) and e(T’) generate nonisomorphic minimal left ideals of the group algebra KS,, whenever T and T‘ belong to different frames associated with n. ( I .21) REMARK. The observation was made in Remark (1.9) that the number of conjugacy classes of S,, is equal to the number of distinct partitions of n. This comes about in the following manner. Each permutation in S, can be written in a unique way,

(1.22)

s = (n,1

. . . n1kJ . . . (%,I

. . . %k,,,),

as the product of n? disjoint cycles (nil ... niki), 1 5 i 5 m. This cycle structure of s is characterized numerically by giving the number a, of cycles of length 1, the number a2 of cycles of length 2, . . . , the number cx, of cycles of length r . Two permutations s and t of S, are conjugate if and only if they have the same cycle structure. The cycle structure of a conjugacy class X of clements of S, can be associated with a partition of n or with a frame F i n a natural fashion. For example, the frame F given by Fig. (1.23) specifies the (1.23)

clas< K of S , , consisting of all permutations with cycle structure, two cycles of length 2, one cycle of length 3, and one cycle of length 4. These results are read from the frame F by starting with the last row of two boxes indicating a ‘-cycle, the next to last row of two boxes indicating a second 2-cycle, the second from last row of three boxes indicating a 3-cycle, and finally the first row of four boxes indicating a 4-cycle. Many writers specify this class by either the symbol 2’3’4’ or 2234. More generally, the symbol

.

1 1 1 , ” i i 7 ~ ~.~. ~7:

with

51,177,

+ . . . + arm, = n

denotes the cla\s K of all permutations of S, with cycle structure: cxl cycles of length l l i l , . . . , x, cycles of length m,.

223

1. The Representation Theory of the Symmetric Group

(1.24) REMARK.The standard formula for the number of elements of the class K is n! m l a l a ,! * . . mP,ar! The number of distinct classes of isomorphic irreducible representations of S, over the field of complex numbers is equal to the number of distinct classes of conjugate elements and thereby to the number of different partitions or frames. As a specific example of the preceding discussion, consider the case of the group S, whose seven distinct classes of equivalent irreducible representations arise from the frames shown in Fig. (1.25). These are arranged in descending (1.25)

order according to the standard ordering convention. Each gives rise to an irreducible matrix representation called a Young’s ir~tegrairepresentarion. Two such integral representations belonging to different frames are not equivalent so that these representations contain a representative element from each class of equivalent irreducible representations of S, . Extending the notation to the corresponding representations, the seven frames of ( I .25) determine the irreducible representations shown in Fig. (1.26), whose ordering (1.26)

T ....., T ...., T . . . , T ..., T . . , T . . , T . .. ..

is taken to be that of the frames. This correspondence between frames and representations and the induced ordering is tacitly understood in the sequel, although it is frequently established by means of an enumeration such as

TI, TZ 3

T3

T,,

T5, T s , T7.

According to Chapter 2, each irreducible representation q,, is acsociated with a simple component J‘.’ of the group algebra KS, . Consequently, the group algebra KS, decomposes as the direct sum

J””’@J’

....

...

... @J” @ J ’

..

.. .. B.7‘ @ J ‘ @ J ’ ,

224

4. Representation Theory of Special Groups

where the notation is introduced in the obvious manner. It is usually more convenient to write the decomposition as

J’ @ J 2 @ J 3 @ J 4 @ J s @ J 6 @ J’.

(1.27) REMARK.The calculation of the dimension of the Young’s integral representation belonging to a given frame is illustrated in the case of the frame

E F

shown in the diagram which belongs to the partition (5, 3, 1, l} of 10. One substitutes for the frame its node diagram comprising the array

***** ***

(1.28)

*

* The principal nodes of a node diagram are those lying in the first column. To each node there corresponds a hook which is the set consisting of the given node together with all other nodes lying in the same row to the right and in the same column below. The length of a hook is the number of nodes it contains. The hook graph is obtained by replacing each node of a diagram by its hook length. The hook graph of the node diagram (1.28)is

85421 521 . 2 1 The dimension f of the irreducible representation T of S, belonging to the frame F is the quotient of n ! by the product of all the hook lengths of the hook graph of F,

(1.29)

f

= n!/(product

of the hook lengths).

In the particular instance of (1.28),one finds

for the dimension of the associated irreducible representation. The concept of hook plays an important role in the representation theory of the symmetric

225

1. The Representation Theory of the Symmetric Group

groups, but it will be beyond the scope of this book to discuss these matters at greater length. The set {f i , . . . ,f,} of the dimensions of the irreducible representations of S5 corresponding to the set of frames (1.25) is {I, 4, 5, 6, 5 , 4, I}, respectively, according to Eq. (1.29). The irreducible representation T . . . corresponding to the frame

..

: :. has dimension five. It follows from the general theory of Chapter 2 ...

that the simple component J

(1.30)

J

... "

'

'

of the group algebra KS, is the direct sum

= L' @ Lz Q L3 Q L4 @ Ls

of minimal left ideals L', 1 5 i < 5. The minimal left ideals in a particular decomposition of J"

can be determined by means of the set of standard

. These are the tableaux

tableaux associated with the frame

(1.31)

123 45

124 35

125 34

134 25

135 24 .

(1.32) DEFINITION. A standard tableau T belonging to the frame F is any tableau belonging to F i n which the numbers in each row increase from left to right and those in each column increase from top to bottom. (1.32') THEOREM.The number of standard tableaux associated with a frame F is the number of minimal left ideals in any direct sum decomposition of the simple component of KS, associated with F. This is the dimension of the irreducible representation of S, belonging to F. Equation (1.30), for example, can be written ... 123 124 125 134 135 J " = L 4 5 @ L 3 5 Q L 3 4 @ L2 5 @ L 2 4 , where each of the minimal left ideals in the decomposition is characterized by the standard tableau with which it is labeled. In any group algebra K S , , a similar decomposition exists for each simple component J. The standard minimal left ideals of J are those corresponding to the standard tableaux of the frame F of J .

(1.33) REMARK. We have accumulated sufficient terminology and results to give a full description of the standard decomposition of the group algebra KS, into its irreducible parts. First, there exists an ordered set {F(l),. . . , F(r)} of frames in one-to-one correspondence with the partitions of n. Second,

226

4. Representation Theory of Special Groups

there exists an ordered set {T(i,I), . . . , T(i,fi)}of standard tableaux associated with each frame F ( i ) , 1 5 i 5 r. The group algebra KS,, is the direct sum KS,

=J F ( I )

@

. . . @JF(*),

where each of the simple components is itself a direct sum

(I 24)

JF") = L T ( i , 1 )

@

...@ Ln', fr)

of standard minimal left idmls. Each standard minimal left ideal of (1.34) is generated by a standard primitive idmipotent arising from a standard tableau. The standard tableaux are ordered according to a scheme deducible from (1.31). The tableau Tiprecedes the tableau T, if, reading from left t o right and top to bottom along the boxes of the frame, the entry in Tiis smaller than the corresponding entry of T, at the first difference. The basic task remaining is a description of the process of computing the matrix of s in the Young's integral representation. Before attempting this, we give two useful schemes for calculating characters and representations of the symmetric groups. (1.34') REMARK. Let K denote any class of conjugate elements of the symmetric group S,, . We outline a computational procedure for determining the value at K of the character of the irreducible representation of S,, determined by the partition {ml, . . . , n i k ) of n. The scheme is highly procedural and involves what we call a box for want of a standard terminology. A box is a symbol of the form [nl, . . . , n k ] ,where the entries n i , 1 i 2 k , are integers. A box has the value zero if any of its entries are negative. The standard box containing the intcgers ( 1 2 ~ ) . I 5 i 5 k , is that one in which the entries appear in nonincreasing order. If p is a permutation converting a nonstandard box into a standard one by permutation of the entries, then the value of the nonstandard box is plus o r minus that of the standard one accordingly a s p is a n even or odd permutation. In particular, the value of a box is zero whenever there is a repetition among its entries. The value of a standard box whose entries are nonnegative integers without repetitions is 1 . The calculation proceeds by allowing cycles to act on boxes. To keep notation to a minimum while still illustrating the general idea, consider the action of a cycle of length 1 1 1 on the box [n I , n2 , n 3 , n4, ns 1. The rule is ???[/7,,

n2.n2.n4.

n,l

=

+

[n1 - m. n 2 , n 3 , n 4 . n,] [ n , ,n2 -m, n 3 , n4 > nsl + [nl. / * 2 . 1 7 3 - m , n4, n,l + [n,,n 2 , n 3 , n4-m, nsl [ n , ,n2 , n 3 1 1 4 , ns - m].

+

.

To be specific, Ict a cycle of length 3 act on the box [6, 4, 21.

3[6.4,2]=[3.4,2]+[6,1,2]+[6,4,-l]=-[4,3,2]-[6,2,1],

227

1. The Representation Theory of the Symmetric Group

where nonstandard boxes are replaced by standard ones and the box [6,4, - 11 of value zero is suppressed. One proceeds according to the following steps, each illustrated by the calculation of the value of the character of S, determined by the partition (4, 3, 2) of 9 for the class K with cycle structure 1 2 32. out the node diagram associated with the given partition, (1) Write ....

namely, : . in the example. (2) Determine the lengths of the principal hooks in the hook diagram, namely,

6*** 4** 2* in the example. (3) Associate with the representation the standard box whose entries are the principal hook lengths, namely, [6,4, 21 in our example. (4) Let the cycles of the cycle structure of K act consecutively on the standard box, beginning with the cycles of greatest length. In our example, this process assumes the form

1 2 32[6, 4, 21 -+

= -+

= +

= -+

+ [6, 1,2] + [6,4, - 11) + [6, 1, 21) 1 2{[0,4,2] + [3, 1,21 + [3,4, -11

1 2 3{[3,4,2] 1 2 3{[3, 4, 21

+ 13, 1,21 + [6, -2,21 + [6, 1, -11)

+

1 2{[0, 4, 21 2[3, 1, 21) ~ 2 , 4 , 2 1 +LO, 2,21+ LO, 4,oi 2[1, 1, 21 2[3, - 1, 21 2[3, I , 01)

+

+

1{2[3, 1,OI) 2[2, 1, 01 2[3,0, 01

+

+

+ 2[3, I , -I].

( 5 ) The character of this representation has the value on K given by the final expression when each box is replaced by its numerical value. In the example, one obtains 2(1) 2(0) 2(0) = 2.

+

+

(I .35) REMARK. The development of the representing matrices in the case of

Young’s integral representations is a tedious process which we sketch below. However, there is a simpler process for obtaining the matrices of Young’s

228

4. Representation Theory of Special Groups

rational seminormal form. We give a summary of the results. For additional details, the reader may consult Robinson (1961). Since every element s E S, can be written as the product of transpositions, it proves sufficient to determine the matrices which represent the transposition (r, r l), 1 I r < n, in Young's rational serninormal form. One has the following fundamental theorem.

+

(1.36) THEOREM. Let F be the frame belonging to the partition {ml, . . . , mk} of n. Let T , , ... , Tfbe the standard tableaux associated with F given in their natural order. The irreducible representation of S, corresponding to F is an f x f'matrix representation. The rows and columns of the matrix representing the transposition (r, r 1) can be labeled by means o f the standard tableaux. The resulting matrix o f t = (r, r 1 ) has the general form shown in Fig. (1.37).

+

+

(1.37)

Tf

The matrix o f t is found by advancing along the principal diagonal. Having reached the position (u, u ) or (T,, T,), entries are determined by the rules:

( I ) r,,, = 1, when r and r + 1 occur in the same row of T, . (2) t,, = - I , when r and r + 1 occur in the same column of Tu. (3) When rand r + 1 occur neither in the same row nor in the same column, the procedure is more complicated and produces off-diagonal as well as diagonal elements. Assume that r and r + 1 are in positions (m, n) and (p, q), respectively, of T, . One examines the standard tableaux following T, for a T, which coincides with T, when the locations o f r and r + 1 are interchanged. Since To follows T,, note that m < p and q < n. When T, is discovered, one takes t,, = -2,

where I/].

= (n

-

t,"

=

1 - I?,

t""

=

1,

t,, = A,

nz) - (q - p ) .

(4) Zeros occur in all other positions of the matrix.

...

As an application of the ideas of (1.36), consider the representation T ' of S , belonging to the partition (3, 1,

... l} of 5. We find the matrix {tij} = T' (t)

229

1. The Representation Theory of the Symmetric Group

where t is the transposition (34). The standard tableaux of the frame F belonging to (3, 1, 1) are 123 Tl=4 ,

124 T2=3 ,

125 T3=3 , 4

134 T4=2 , 5

135 T5=2 , 4

145 T6=2 . 3

5

5

The final result is presented in Fig. (1.38). We begin the evaluation of the element t , , by noting that 3 and 4 occur neither in the same row nor the same column of TI. However, T, results from the transposition of the elements 3 and 4 in T, . The element 3 is in the position (1, 3) = (m, n) of TI, while the element 4 is in the position (2, 1) = ( p , 4). Thus one finds I/A = (3 - 1) - (1 - 2 ) = 3. Hence, we have t , , = -1/3, t I 2 = 819, t , , = 1, and t,, = 113 which completes the first two rows and columns of the matrix. Continuing with t,, , we see that 3 and 4 are in the same column of T3which gives t 3 3 = - 1 and completes the third row and third column. Now, for t,, , we note that 3 and 4 are in the same row of T, which gives t44 = 1, completing the fourth row and fourth column. Finally, T5is obtained from T6 by the interchange of 3 and 4. The element 3 is in position (1, 2) = ( m , n), and the element 4 is in the position (3, 1) = ( p , 4 ) . This gives l/A = (2 - 1) - (1 - 3 ) = 3. Consequently, the nonzero elements of the last two rows and columns are: t,, = - 1/3, t,, = 8/9, t , , = 1, and t66 = 1/3. This completes the work and gives the result shown in Fig. 1.38. (1.38)

I -1/3

S/9

1 1 / 3 0 0 0 0 0 0 0 0

1

0 0

0 0 0 1 0 0

0 0

0

0 0 0 0 0 0 0 0 -1/3 8/9 1 1 / 3

We turn now to an outline of the calculation of the matrix of s E S, in the Young’s integral representation determined by some frame F belonging to n. We observed without full proof that the set {TJ,1 < i sf, of standard tableaux belonging to F determine a corresponding set {e(T,)) of essential idempotents such that the simple component J F of the group algebra A of S, is the direct sum of minimal left ideals

J F = Ae(T,)

+

*

A

-

+ Ae(Tf).

230

4. Representation Theory of’Special Groups

Difficulties arise from the fact that the idempotents corresponding to the set te(Ti))do not form a complete set of orthogonal primitive idempotents of J r . One has only the following lemma. (1.39) LEMMA.The product e(Ti)e(Tj)= 0 whenever Tiand < Ti. diagrams belonging to a frame F with

TJ

are standard

The proof is omitted. Unfortunately. the product e(Tj)e(Ti)need not be zero under the conditions of the lemma. Since a straightforward calculation of the matrix corresponding to ;I group element .r requires a complete set of orthogonal primitive idempotents, it becomes necessary to make a troublesome transformation on the set {(>(Ti)]. I I i if.Notwithstanding, it is possible to efrectively define a subset CZ’ of the group algebra A consisting of quantities {N,~, , wr} such that the set E of elements ( e l , . . . , e,} defined by (1.40)

ei = e(Ti)wi,

1I i cJ;

is a complete set of orthogonal idempotents for the minimal two-sided ideal J F. The matrix { s i j ] ,I <: i, , j Cf, associated with s by means of the set E is determined by (1.41)

eisej= s i j e i j ,

1I i,,j

is a set of matrix units associated with E. where {eij},1 i i , j Boerner establishes all of the details of an algorithm for the calculation of the . Y ~ We ~ restrict . ourselves to an outline of the algorithm and t o an illustration of some of the details. Unfortunately, a really good example needs to be of such high dimension that we are forced to use some less satisfactory ones of lower dimensions. The principal steps are as follows: (1) Write down the sequence of standard tableaux TI, ..., Tf corresponding to a given frame F in increasing standard order. This has been carried out in Fig. (1.42) for the frame F whose hook diagram is

53 1 31 1 One checks that the dimension f of the corresponding irreducible representation is 6!/(1(3(3(5)))) = 16. ( 2 ) Beginning with T , and continuing through T,-,, one determines thoae T, greater than Tisuch that some pair collinear in T, is cocolumnar in T, . This process has been carried out for the Lableaux T , , T, , TI I , T,2 , T,3 , arid T,, with the results tabulated in rows TI through T14of the Fig. (1.42).

231

1. The Representation Theory of the Symmetric Group

In the case of TI, one finds that each of the diagrams TI, TI], and T12has no collinear pair (a, b) which is cocolumnar in TI.A collinear pair (a, b) in lj greater than TI which is cocolumnar in TI is listed in row TI under column T i . The remaining rows supply the same information for T, through T14. Given a pair T i , 7', with Ti < q-and no pair (a, b) cocolumnar in 17;. and collinear in T j , there exists a p i j E P(T,.) such that each column of Ti contains (1.42) Ti 123 45 6 Ti 123 Ti 45 6 T2 123 45 46 5 TI1 135 12 24 6 Tlz 135 12 26 4 T13 136 12 24 5 TI4 136 12 25 4

MATRIXPREPARATION Tz 123 46 5 46

T3 124 35 6 14

T4 124 36 5 14

Ts 125 34 6 25

Ts 125 36 4 25

Tz

14

14

15

15

26

26

14

14

15

12

12

12

12

12

12

12

34

26

Ti1 26

12

12

12

12

12

12

12

14

14

24

Tlz 24

12

12

12

12

12

12

12

25

34

12

12

12

12

12

12

12

14

14

T7 Ts Ts Ti, Tii Ti, 126 126 134 134 135 135 34 35 25 26 24 26 5 4 6 5 6 4 16 16 14 14 Ti1 Ti2

Ti, Ti4 Ti5

Ti6

136 24 5 16

136 25 4 16

145 26 3 14

146 25 3 14

Ti3

7'14

14

14

16

16 26

16

36

14

14

15

15 TI3 25

15

25

35

35

T14 14

14

15

24

the same elements as the corresponding column of p i j T , . One proceeds through the list of standard tableaux constructing a list of such p i j . In the case of Fig. (1.42), the list begins with p l , l l and p l , l z ,where

(3) Using this list, one constructs for each standard tableau TI a list of all possible permutations {ri,in}of the form ri,i" ' p .[ , t.i p i. i . , .~ . . Pi,,-I,in, (1.43)

where i < i, < . . . < (1.44)

< in. With each such r i , i nthere is associated a sign Ei'" = (-

1>",

where n is the number of factors in r i , i n .

232

4. Representation Theory of Special Groups

(4) Let s E S,, . T o determine the elements of the ith row of the matrix of s under the given representation, one constructs from the set {Ti, . . ., ri,i,} of the permutations r with initial index i, a second set {,,

(1.45)

{s, r i , i

. . ., r i , iks)

, ~ ,

of permutations depending on s. Then one computes the tableaux T'O'

= $-I

T i , . . ., Fk'= ( T i , i k s ) - l T i .

As an illustration, consider the element s = (234) E S , . In order t o compute the matrix element s l m , 1 5 rn i 16, one selects the permutations p l , l = (24)(35) a n d p , , , , = (26)(35). Then one calculatesp,,,,s = (253) a n d p , , , , s = (25346). The list of permutations forming the set (1.45) is (1.45')

f(234), (253), (25346)).

One obtains TCO) =

123 142 (243)45 = 35 , 6 6

T ( ' )=

123 135 (235)45 = 42 , 6 6

T"'

123

= (26435)45

6

165

= 32

4

for the associated tableaux. (5) Each of the tableaux T'O', T ( ' ) , ..., T ( k )makes a contribution E(". 0 5 j . k , to each s i n l ,1 5 rn If,whose value is given by $. I

", =

+ c(l) + ... + E(k),

where the c's are understood to depend on the column index rn of s i m . Each T('.' is related to an r i , with an associated sign q(',)= (- 1)"" where n, is the number of factors of r i ,,. The r corresponding to T'O) is the identity and the associated sign plus. m <,f,if T(') has a cocolumnar pair (a, 6)which (5a) For a given s i m , I I is collinear in T,, then

,

( I .46)

p)= 0.

(5b) If T(" has no cocolumnar pair (a, b ) which is collinear in T,, then there exists q E Q(T('))such that each row of qT('.)contains the same elements

233

1. The Representation Theory of the Symmetric Group

as the corresponding row of T, . Let cq = I for q even and one has

-

1 for q odd. Then

p)= Eq Fi(L).

(1.47)

The application of these rules to obtain s,,,, , 1 I m 5 16, is carried out in Table (1.48). The tableaux T(O),T(’),and T(’) are listed at the side of the

(1.48) TI

MATRIX ENTRIES FROM TABLEAUX

Tz

T3

T4

Ts

T6

T7

T8

T,

Tlo Ti1 Tiz

Ti3

Ti4 Ti,

Ti6

123 123 124 124 125 125 126 126 134 134 135 135 136 136 145 146 45 46 35 36 34 36 34 35 25 26 24 26 24 25 26 25 6 5 6 5 6 4 5 4 6 5 6 4 5 4 3 3 142 (13) (13) 1 35 1 6

(36) -1 -1

(36) (16) (16) (13) (13) (13) (13) (13) (13) (45) (16)

135 (23) (23) (14) (14) -1 1 (16) (16) (14) (14) 1 -1 (16) (16) (14) (14) 42 1 -1 -1 1 6 165 (13) (13) (14) (14) (34) -1 (26) (26) (14) (14) (13) (13) (13) (13) (14) (14) 32 1 4 s1,=0

0

1

0

0

0

0

0

0

0

-11

0

0

0

0

table. When T, contains a collinear pair (a, b) which is co-columnar in one of the tableaux T(’), T(’),or T(’), this pair is listed under T,, in the corresponding row. Otherwise, the value of c4 is listed followed by that of E ( ” ) = E ~ E ~ ( ’ )This . second type of listing first occurs in row

142 T(O)= 35 6 under T, = T 3 ,where cq involved is q = ( I ) since

=

I and

=

I , so that do)= 1 for m

I42 35 6 has the same rows as

124 T3 = 35 6

= 3.

The q

234

4. Representation Theory of Special Groups

One notes in general that = - 1, and E , ( ~ = ) - 1 in our example. = 1, The values of I 5 nz 5 16, are listed in the last row of Table (1.48). (1.49) EXAMPLE. The reader may wish to check his understanding of the procedure by determining the matrix of (234) in the irreducible representation of S , given by the frame : : '. The result should be as shown in Table (1.50).

( I .50)

123 124 125 134 135 45 35 34 25 24 123 45

0

1

0

0

-1

124 35

0

0

0

1

-1

125 34

0

0

0

0

-1

134 25

1

0

0

0

-1

135

0

0

1

0

-1

I t is clear that the calculations involved are rather tedious for even small values of n. P. D. Swardstroni has developed a program operating on the 1 BM 7094 to make these computations for the representations of S,, , 3 1.17 5 7. The method works in general but limitations of machine storage make it impractical to deal with larger values of n or even with some of the higherdimensional representations of S , and S , . We turn now to a discussion of the relations between the representations of the symmetric group S,, and those of the general linear groups GL(V) of nonsingular, linear transformations on an rn-dimensional vector space.

2. MODULES OVER SYMMETRIC ALGEBRAS

This section defines the concept of a symmetric algebra and investigates the special properties of a' module M over a symmetric algebra A. The word algebra always refers to an algebra A with identity 1 which is finite-dimensional over the complex numbers. The treatment is confined largely t o the case where the symmetric algebra A is the group algebra KS, over the complex

2. Modules over Symmetric Algebras

235

field of the symmetric group S,. These considerations lead naturally to a presentation of the elegant method of Curtis (1956, 1958) for establishing the relationships between the ideal theory of KS, and the integral representations of the general linear group GL(V) of nonsingular linear transformations on a complex, finite-dimensional space V. The treatment of this section admits of a substantial generalization, but we must refer the reader to the papers of Curtis or to the monograph of Curtis and Reiner for additional information. We state the definition of semisimplicity in the case of a finite-dimensional algebra A with identity. Then give the version of Wedderburn's theorem which holds in this special case. Every semisimple algebra is shown to be a symmetric algebra. Given any module M over the symmetric algebra A, there exists a particular two-sided ideal A, of A which is called the nucleus of M. Let D denote the algebra HomA(M, M) acting on M as right-multiplication. Whenever the nucleus AM has an identity, there exists a natural duality between the right-ideal structure of AM and the right D-components of M. Let M denote the n-fold tensor product V 0 . .. @ V of the m-dimensional, complex space V with itself. Then M can be made into a KS,-module. Furthermore, M affords a tensor representation T of the general linear group CL(V). Let A denote the complex group algebra KS, of the symmetric group S, and b the enveloping algebra of the group T(GL(V))of linear transformations on M. Then A is a symmetric algebra on M and b is Hom,(M, M). Thus one can apply the general theory to deduce the relations between the ideal theory of KS, and the tensor representations of GL(V). We recall some definitions from Chapter 2. Let A be a finite-dimensional algebra over the field K of complex numbers. An ideal J of A is said to be nilpotent if there exists a positive integer n such that the product of n or more factors from J is always 0. The sum of all the nilpotent left ideals of the algebra A is a nilpotent, two-sided ideal R called the radical of A. A finitedimensional algebra A with identity 1 is said to be semisimple if and only if A contains no nonzero, nilpotent two-sided ideals. A semisimple algebra A which contains no nontrivial two-sided ideals is called simple. There is a famous theorem of Wedderburn which can be stated in the case at hand as follows. (2.1) THEOREM (Wedderburn). Let A be an algebra with identity such that A is finite-dimensional over the complex numbers K. Suppose that A is semisimple. Then A is the direct sum (2.2)

A = J1@...@J'

of minimal two-sided ideals J', I i i r , such that each J' is a simple algebra over K . Furthermore, any minimal two-sided ideal J of A coincides with one of the J'. Finally, there exist integers n, such that each J', 1 i i i r, is iso-

236

4. Representation Theory of Special Groups

morphic to the full matrix algebra of all n, x n , matrices over the complex numbers. As in Chapter 2, we use the symbol

B=B(l)@*..@B(r) for the set of all quasi-diagonal matrices of the form

where each B(i) denotes the algebra of all n , x n i complex matrices, 1 5 i 5 r. The set B is a subalgebra of the algebra of all LY x LY complex matrices where LY = n,

+ ... + n,.

By Wedderburn’s theorem, there is a family of homomorphisms hi : A such that Im h i = B(i). The map h : A B defined by

-+

B(i)

--f

(2.4)

h(x) = h,(x)

+ . . + hr(x) ’

is an isomorphism of A onto B.

(2.5) DEFINITION. A finite-dimensional algebra A over the complex numbers K is said to be symmetric if and only if there exists a nondegenerate bilinear formf: A x A -+ K such that f(a, b)

(2.6)

=f@, a)

and

(2.7) for all a, b, c in A. A bilinear form satisfying (2.6) is symmetric and one satisfying (2.7) is associative. (2.8) REMARK.Every finite-dimensional, semisimple algebra A over the complex numbers K js a symmetric algebra. To see this, let h : A -+ B be the isomorphism (2.4) defined previously. For a, b E A, one has

h(a)

= x1

+ ... + xr,

xi € B i ,

and h(b)=y, + * - * + y , , Then define f(a, b)

= tr(x,y,)

yi€Bi.

+ . . + tr(x, y J *

237

2. Modules over Symmetric Algebras

+

+

One has tr((x y)z) = tr(xz yz) = tr(xz) + tr(yz) for matrices x, y, and z , so thatf(a b, c) =f(a, c) +f(b, c). Similarly,f(a, b c) =f(a, b) +f(a, c). Also tr((crx)y) = tr(x(ay)) = a tr(xy) for matrices x and y and complex numbers a so that f is a bilinear form on A. It follows also from tr(xy) = tr(yx) thatf(a, b) =f(b, a). Since tr((xy)z) = tr(x(yz)) for matrices x, y , and z, f(ab, c) =f(a, bc), which proves thatfis associative. Furthermore. given any a E A, there exists an a* E A such that h(a*) = x,* + ... + x,*, where xi* denotes the matrix Hilbert adjoint to x i . Thus

+

f(a, a*) = tr(x,x,*)

+

+ . . . + tr(x,x,*).

Recall that tr(xixi*) is positive unless xi = xi* = 0. It follows that either f(a, A) = 0 orf(A, a) = 0 implies that a = 0, so thatfis nondegenerate. Consequently, A is a symmetric algebra. We see, i n particular, that KG is a symmetric algebra for every finite group G. Every choice of a basis {a,}, 1 5 i I n, of a complex n-dimensional algebra A gives rise to two matrix representations of A. The first of them is defined by x -+ A(x), where xa, = 2 ,I(X)~,aj,

and the second by x

+ A(x),

a,x

=

1I iI n,

where

1A(x)ijaj,

1I iI n.

In the case of a symmetric algebra .4, there exists a nice relationship between these representations of A determined by the basis {a,} and those similarly defined by the dual basis (b,} of (a,} with respect to the associative formf. (2.9) LEMMA. Let {a,}, 1 5 i 5 n, be a basis of the symmetric algebra A and let {bi}, 1 5 i 5 n , be its dual with respect to the formf of A, that is,

f ( a i , bj) =f(bj, a,) = d i j . Then, for 1 5 i 5 n , one has the equations, xai =

A(x)~, a j ,

xb, = Proof. First, note that

1A(X)~,bj ,

(2.10)

a, x

=

iff (2.1 1)

bi x =

A(x),~a j ,

c A ( x ) ~bj~.

c , I ( x ) ~aj~ x , f ( x a , , bj)aj, so that A(x)~,=f(xai, bj). Moreover, one has b, x c f(bi x, aj)bj c f(bi, xaj)b, c f(xaj, bi)bj xai =

=

=

The remainder is analogous.

=

=

=

2(x),, bj .

238

4. Representution The0 ry of' Special Groups

Let M be a finite-dimensional complex vector space which is a left A-module of the algebra A.

(2.12) INiwITiox. The dual space M" of M is a right A-module with multiplication of,/'€ M" by a E .I defined by ( fa)x =f(ax),

x E M.

7 he pr,mf that R.1" is ii right A-module with this definition of multiplication is

i t 3 to the reader. Our goal is to introduce a certain two-sided ideal A, of A, called the nucleus of M. which plays a significant role i n developing the integral representation theory of the general linear group GL(V). The nucleus is defined with the aid of ii map 7 : M x M" -+A which we now introduce. Let {a,} and {bi]. denote a pair of bases of the symmetric algebra A dual with respect to the associative bilinear formf'of A. Given (m,f) E M x iM*, define y by y(nl7.f) =

C biJ(ai m).

I t is easy to verify that y is bilinear over K. In addition, y is bilinear over A in the sense that

?(am1

+ bm, ? . f=) ay(m,,.f) + by(m2 , f )

and y(m. /;a +.fz b)

where m. m , , m2 E M,.f: ./;. ,f2 argument i s to show that y(am,f') m E M , ~ M". E and a aJld (2.11).

E

=

E

=

IJ(m.f1)a + Y(m?.fz)b,

M* and a, b

a:;(m, f ) .

E

y(m,fa)

A. The difficult part of the =

r(m,.f)a,

A . We demonstrate the first of these using Eqs. (2.10)

~ ( a m - f )= C bi.f(ai(am)) = bi./((aia>m) = b ; , f ( ( CA(a);jaj)m) = 1 A(a),j b,f'(ajm) = 1 abjf(ajm) = a bjf(aj m) = ay(m,f).

C

c

I3

The argument that y(m,./a) = y(m.f')a is similar. The form y is nondegenerate. For suppose that y(m,f) = 0 for every f ' M*. ~ Then 0 = bif'(a,m) implies that f(aim) = 0 for all i. Since the identity of A is a complex linear combination of the basis {a,), this means that.f(m) == 0 for a l 1 . f ' ~M* so that m itself must be the zero of M. We leave to the reader the argument that y(m.,f)= 0 for all m E M implies t h a t f = 0. The bilincarity of 7 in this second sense implies that the linear span of all elements of A of the form y ( r n . f ) , m E M, f E M*, is a two-sided ideal J of A.

1

239

2. Modules over Symmetric Algebras

The ideal J defined above is called the nucleus of M and (2.13) DEFINITION. denoted by AM. Since our analysis is restricted to the case in which A is a semisimple algebra, one has

A=A,oA,

(2.14)

where A is a two-sided ideal complementary to A,. According to (2.14), the identity 1 of A decomposes as the sum 1’ + I”, 1’ E AM, 1” E A, of orthogonal central idempotents. In particular, A, is generated by the central idempotent 1’ which acts as the identity of AM. The group algebra FG of a finite group G over a field F of characteristic p , p dividing [G : I J, is an exarnple of a symmetric algebra which is not semisimple. In the case of such more general symmetric algebras, a left A-module M is called regular exactly in those cases where A, has an identity. The identity 1’ of A, can be written in the form

1’ = C y(mi

(2.15)

where mi E M, f , E M*, 1 5 i 2 k . For f E M*, one has that y(m,f-fl’) y(m,f) - y(m,f)l’ = 0 for all m in M which implies that

=

f l , ’ =fl’ = f :

(2.16)

Thus the right-multiplication 1,’ acts as the identity on M*. Similarly, the left-multiplication 1,’ acts as the identity on M, that is,

1L ’m = l’m = m,

(2.17)

m E M.

(2.18) LEMMA.The ideal A is the kernel of the representation of A afforded by M while AM is faithfully represented on M. Proof. Let x be an element of the kernel K of the representation of A afforded by M. Then XI’ = x

C ?(mi ,fi)= C y(xmi ,fi) = 0

so that K c A.Conversely, let x E

A.Then

xm = x(1’m)

= (x1’)m = 0

for every m E M so that A c K. Since AM n A = 0, AM is faithfully represented on M. The set D = Hom,(M, M) is a complex algebra playing a central role in our later analysis. We do not verify that D is an algebra, but observe that if u, v E D, then u v is a homomorphism of M which clearly commutes with multiplication by elements of A. A similar statement can be made about the product uv. Details are left to the reader as an exercise. Since A has an identity

+

240

4. Representation Theory of Special Groups

1 and therefore contains K , D is a subalgebra of Hom,(M, M). We have the following useful theorem.

(2.19) THEOREM. Let M be a left A-module for the semisimple algebra A over the complex numbers K . Then D = Hom,(M, M) is a finite-dimensional, semisimple algebra over the complex numbers. Proof First observe that HomA(M, M) = HomAM(M,M) since A is the kernel of the representation. The nucleus AM is a semisimple K-algebra faithfully represented on M. Hence there exists a K-basis B = {ml, . . . , m,} unitary with respect to a suitable inner product of M, such that each lefttranslation aL, a E AM, has a quasi-diagonal matrix with respect to B. Denote it by

where each of the Ai’s is a square matrix of some fixed dimension n, x n i , independent of the choice of a E AM.Given any matrix M(a,), there exists a* in AMsuch that M(aL*)= M(a,)*. From a,*d = daL*,one obtains

or (2.20)

M(a,)*M(d)

=

M(d)M(aL)*

M(d)*M(a,)

= M(a,)M(d)*.

The linear transformation d* with matrix M(d)* satisfies

d*aL= a,d*, which implies that d* E D. Thus D is closed under the adjoint operation. Let d belong to a nilpotent, two-sided ideal of D. Since d*d is nilpotent, tr(d*d) = 0, which implies that d = 0. Therefore D is semisimple since it contains no nontrivial, nilpotent, two-sided ideals. Finally, D is finite-dimensional over K as it is a subalgebra of the finite-dimensional algebra Hom,(M, M). The study of the various relationships between D and AM is facilitated by the introduction of a map f l x I y : M + M , corresponding to each pair, J’E M*, y E M, which is defined for m E M by

m(fl x I Y> = YhS)Y. The functionfl x 1 y is an element of D = Hom,(M, M). First note that

(am>(Slx I Y> = r(amJ)Y = [ar(m,f)lu = a[r(mJ>yl= “(fl x I Y)1 ~ y, m E M, and a E A. Consequently, f ]x y commutes with for , j ’ M*, multiplication by a. The remainder of the argument is left to the reader.

241

2. Modules over Symmetric Algebras

(2.21) THEOREM.For each t E Hom,(M, M) there exists a t = aL. Proof. Given t E Hom,(M, M), it follows that

Wfl x I Y)1 Let a = y(tmi,fi), where 1' = m E M, one has

=r(mf)y,

E

A such that

m E M.

y(mi,f,) is given by Eq. (2.15). Then, for

aLm = am = [C ~(tmi,fi)]m = C r(fmi,fi>m = C t[y(mi ,fi>mI = t[C y(mi >fi>m]= tm.

Hence, t

= aL,as

was to be shown.

The custom is t o write the elements of A to the left of the elements of M on which they operate and those of D to the right. The commutativity between A and D = Hom,(M, M ) is then expressed by (am)d = a(md) for m E M, a E A, and d E D. It is indispensible to our methods to establish a one-to-one correspondence between the right ideals of the form eA, with e an idempotent of A,, and the right D-components of M. Since D is semisimple, every right D-submodule N of M is a D-component of M. Let J denote the set of right ideals of A, and '3 the set of right D-components of M. Define the map A: J % by A(eA) = eM = eAM. Then eM is a right D-submodule since (eM)d = e(Md) for d E D. If e and f a r e idempotents of A, with e A = fA, then fe = e and ef = f so that e M c f M and f M c e M . Thus A is well defined. Define the map p : % + J for any right D-submodule N of M by p(N) is the subspace of A, generated by all elements of the form y(n, m*), n E N, m* EM*. Since y(n, m*)a = y(n, m*a) for a E A, p(N) is a right ideal of A,. --f

(2.22) LEMMA. The map A: J + '3 is a bijection of J onto '3 with inverse the map p : '3+ J, a bijection of % onto 3. Proof. Let the right ideal fA of A, contain the right ideal eA, with f and e idempotents of A,, so that fe = e. Consequently, A(eA) = eM c f M = A(fA). Let the right D-submodule N of M contain the right D-submodule P. Then each element y(p, m*) generated by P is also generated by N. Therefore, p(P) c p(N). Thus both A and p are inclusion preserving maps. lf x E eM, then y(x, m*) = y(ex, m*) = ey(x, m*),

m* E M*, is an element of eA which implies that p(eM) c eA. Conversely, let x E e A c A M .Then

x = XI'

=x

C r ( m i > L >= C y(xmi,fi) = 1 y(exm, ,si)

242

4. Representation Theory of Special Groups

is an element of p(eM). This result shows that eA c p(eM) Hence, ,d.(eA)

= p(A(eA)) ==

= p(I.(eA))

c eA.

eA and p?is the identity map on J.

Let N be a right D-component of M. There exists a projection 71 E

Hom,(M, M)

of Nl onto N since D is semisimple. By Theorem (2.21), there exists an e E A, with mn = em. m E M. I t follows that (e2 - e)M = 0, which means that e2 = e since A, is faithfully represented on M. Note also that y(n, pi*) = y(en, m*)= ey(n, m*),

n

E

N, m*

E

M", so that p(N) is contained in eA, a right ideal of A,. However, e

= el' = e

2y(mi ,f;)= 2 y(emi ,.fi)

is an element of p(N) so that eA c p(N). Furthermore, 2(,u(N))= 3.(eA)

= eM = N.

Thus j.p is the identity map on %. We conclude that both ;Iand ,D are bijections which are the inverses of each other. We are able to show that 1 and p also preserve algebraic structure. The first result in this direction is the following theorem. (2.23) THEOREM. Let the right ideal eA of A, be A-isomorphic to the right ideal fA of A,. Then the right D-submodule R(eA) is D-isomorphic to the right D-submodule i(fA). Proof: Let / I be an A-isomorphism of the right ideal eA onto the right ideal fA. Then, for x E eA,

h(x) where a

=

h(e)

E

= h(ex) = h(e)x = ax,

fA. Also, for x

E fA,

K'(X) = h-'(fx) where b

=

K'(f)

E

= h-'(f)x = bx,

eA. I t follows that

(ba)x = x, x E ~ A ,

and

(ab)x

=

x,

xE~A.

We define two maps, cr: eM + f M (i(eA) -+ A(fA)) and z: f M - + e M (;I(fA) + i(eA)), by o(x)

ax

E

fM.

x

E

1(eA) = eM,

s(x) = bx

E

eM,

x

E

A(fA)

=

and = fM.

243

2. Modules ouer Symmetric Algebras

These two maps are easily seen to be D-homomorphisms, A(eA) + A(fA) and A(fA) + 1(eA), respectively. Note that (za)(em) = (ba)em = ((ba)e)m

= em

and (az)(fm) = (ab)fm so that zu = l l ( e A ) and az = which completes the argument.

= ((ab)f)m = fm,

Thus a: A(eA) + A(fA) is a D-isomorphism

The companion theorem is also valid. (2.24) THEOREM. Let the right D-submodule N of M be D-isomorphic to the right D-submodule P of M. Then the right ideal p(N) of AM is A-isomorphic to the right ideal p(P) of AM. Proof. Let h : N + P be a D-isomorphism of N onto P and let y(ni', f i ' ) be the zero element of p(N). Then one has, for m E M,

C

(C y(hni',fi'))m

= =

C

C r(hni',fi')m = 1 (hn,Xfi' I x I m)

C h(ni'(fi' I x I m)) C h(y(ni',fi')m)

= h(C

=

y(ni',.fi')m) = h((C y(ni',fi'))m) = 0,

so that y(hni',fi') = 0 since A,is faithfully represented on M. Consequently, there exists a well-defined map h : p(N) + p(P) such that

h"(1 y(ni , f i > ) = C y(hni , f i ) .

The map

h" is easily seen to be additive and, for a E A,

h"((CY(ni >fi>)a)= h"(CY(ni f i a)) = y(hni f i a> = C y(hni L>a= (1 y(hni A'>). =(h"(Cy(ni,fi)))a, >

9

>

3

h" is an A-homomorphism. Similarly, there exists an A-homomorphism : p(P) -+ p(N) defined by

so that

h"-'

h"-'(C?(Pi >A>)= C y(h-'pi

One can see without difficulty that h"-'h" is an A-isomorphism of p(N) onto p(P).

=

--

lp(N)and hh-'

=

so that h

We summarize our results before passing to an important application. We have shown that if A is a complex, semisimple algebra acting on a left Amodule M, then D = HomA(M,M) is a complex semisimple algebra acting on M as a right D-module. Furthermore, there exists a bijection 1from the right ideals of the nucleus AM of M onto the right D-submodules of M such

244

4. Representation Theory of'Special Groups

that 3, maps minimal right ideals of A, onto irreducible right D-submodules of M. Also that eA and fA are A-isomorphic right ideals of A, if and only if /I(eA) and 2(fA) are D-isomorphic right D-submodules of M. These facts provide a powerful tool for analyzing a large class of irreducible representations of the full linear group GL(V) of nonsingular linear transformations on an nz-dimensional complex vector space V. The role of A is played by the group algebra K S , of the symmetric group S, and that of the . @ V of the vector left A-module M by the n-fold tensor product V 0.. space V with itself. We must introduce the group algebra KS, as left operators on the tensor product space M. It is sufficient, of course, to specify how the elements of S, are to act on the basic tensors of M. The action of s E S, is defined by permutation on the positions of the factors of a basic tensor of M rather than by permutation on the elements of a basis of V. Our problem is complicated by the necessity of employing a heavy burden of indices and notational conventions in the sequel. In order to clarify the fundamental idea without introducing superfluous details, we consider first the case of M the four-fold tensor product of V with itself, s = ( 1 4 2 3 ) a n d r = ( 1 4 3 ) , a n d m = x @ w @ z @ y a basic t e n s o r o f M . T h e successive factors of m have been placed out of their natural alphabetical order deliberately. The action of s = (1423) on m to produce sm is given by (1423)[x o

w oz@ y ] = z @ y o w o

x

according to the rules : 3 -+1 under s so that the third factor z of m goes into the first position (a) of sm; (b) 4 -+ 2 under s so that the fourth factor y of m goes into the second position of sm; (c) 2 + 3 under s so that the second factor w of m goes into the third position of sm; (d) 1 -+ 4 under s so that the 1st factor x of m goes into the fourth position of sm. The permutation s = (1423) can be written as

.=( 1

)=(

3 4 2 1 2 3 4

s - q l ) sF'(2) 1 2

s-'(3)

3

s-y))2

which indicates the action of s on m in a manner usefully employed in our general definition below. To check that he has the procedure clearly in mind, the reader may wish to verify that (143) and rs = (1342) have the actions (143)[2 0y 0w

x] = w 0y @ x 0z

245

2. Modules over Symmetric Algebras

and (1342)[x O

w O z Oy] = w O y Ox Oz.

These results show that the action of (1423) followed by that of (143) is the same as the action of (143)(1423). Now we give the general definition in the case where M is the n-fold tensor product of V with itself and s is an arbitrary element of the symmetric group S, . The element s can be written in either of two equivalent forms I (SU)

2 42)

...

n s(n)

or

(s-i(I)

s-’(2) 2

...

s - ’n( n ) ) .

The second of these is more convenient for our immediate purposes, as we have seen previously. Let m denote the basic tensor v, . . . 0v, of M. Then one defines sm by s[v* 0.. . OV,] = vs-l(l) 0 . .. OV,- l ( , ) .

Observe that this is merely a continuation of our initial procedure. We must verify that if r and s are elements of S,, then the action o f s followed by that of r coincides with that of rs. Note that r(sm) is (2.25)

r[v,-l(l) O . . . Ovs-l(,,l

o... Ovs-l(,-I,,),

- vs-l(r-l(l))

) O . . . 0V(rs)- ~ ( n ) 0.. . 0 v,] = (rs)m.

= v(rs)- ‘ ( 1 = rs[vl

One stares at Eq. (2.25) until it can be believed, or else reasons that the entry in the first position of the result r(sm) comes from the r - ’ ( l ) position of sm so that it is v s - l ( r - l ( l ) )the ; entry in the second position of r(sm) comes from the r-’(2) position of sm so that it is v ~ - ~ ( ~ - ~and ( ~ )so ) ; on. This argument establishes the desired fact that the action of S, on a basic tensor satisfies the module requirement, r(sm) = (rs)m. Given any basis { v ~ ., . ., vm} of V, there exists a corresponding extended basis of the n-fold tensor product M consisting of the set {vj, 0. . . 8 vj,f of all basic tensors formed from the basis {vl, . . . , v,} with each indexji, 1 2 i n, ranging independently through the integers from 1 through m. Since we must deal repeatedly with index sets ( j l , . . . ,j,), it becomes convenient to adopt several different shorthand conventions for them. We sometimes use the single symbol (J) to denote such an index set. A general tensor m of M can be expanded in terms of the extended basis as (2.26)

m=

C m ( j l , . . . ,i,)[vj, o ... @vj,],

246

4. Representation Theory of Special Groups

where we maintain our policy of omitting ranges of summation unless there is acute danger of misunderstanding. Equation (2.26) can also be written

m

(2.27)

=

C m(J)v(J).

Thus we use either of the symbols m ( j , , . . . , j n )or m ( J ) to denote the coefficient of a basic tensor itself denoted either by vjl @ . . . @vj,, or v ( J ) . The first o f these, of course, is more specific, but the second more convenient. Each element s E S,, determines a linear transformation T(s) on M whose action on the elements of the extended basis is defined by T(S)[Vjl0. . . O Vj,]

= SIVjl

O . . . 0Vj,I

- ~ j ~ - 1 ( ~ )

0.'. Ovjs-l(,,).

One observes that since the element s acts as a permutation on the elements of the extended basis, the linear transformation T(s) maps a basis onto a basis and is therefore a nonsingular linear transformation on M. One notes as well that if v ( J ) is any element of the extended basis, then T ( ~ s ) v ( J= ) ( ~ s ) v ( J=) r(sv(J))= T ( s ) [ T ( r ) v ( J )= ] [T(r)T(s)]v(J). This equality implies that T(rs) = T(r)T(s) and that the correspondence s + T ( s )is a representation of the symmetric group S, on the tensor product space M. One must always keep in mind that this particular representation is not s 4s 0. . . 0s, n-factors, where s acts as a permutation on the basis vectors of V. To the contrary, we repeat that s acts like a permutation on the set of elements of the extended basis. It follows that the matrix of T(s) with respect to this particular basis is a permutation matrix on the coefficients of a given tensor m of M. We adopt the usual convention and refer to the coefficients of m with respect to the extended basis as the components or tensor components of m. The theory which we elaborate is really nothing more than a systematic analysis of the permutation effected by T ( s )on these components. We begin this analysis in the next paragraph. Let the tensor m of M have the expansion m = m(J)v(J).Then T(s)m is given by T ( s ) C m ( j , ,..',j,,,"j] O . . . O V j " 1

1m(.i,, . . . ,.i,,)s[vj o . . . o vjnl = C m(.j,,. . . ,,jn)[vjs- o . . . o vj,- ](")I. =

I

This expansion can be written in a more suitable form by making a change of indices: k i= j s - l ( i or ) k,(;)=,js-l(s(i)) =ji.This substitution leads to

1

Vs)m = m(k,(,) , . . . , ks(n))[vkl O ... O vk,,I. T o summarize. let us agree that when ( J ) denotes ( , j , , . . . , j n ) ,(sJ) denotes ( j S ( , ).,. . , ,is(,,)). For example, let ( J ) = ( j , , j , , j , , j,) and s = (1342). Then

(2.28)

247

2. Modules over Symmetric Algebras

(sJ)denotes ( j , , j l ,j , ,j2).Equation (2.28) asserts that given a tensor m of M, the components of the tensor T(s)m are related to the components of m by the formula: (2.29)

[T(s)rn](J) = rn(SJ).

Naturally, one usually considers M an S,,-module and frequently prefers to write sm rather than T(s)m. In such cases, Formula (2.29) becomes

sm(J) = m(sJ),

(2.30)

with the interpretation of the symbol (sJ)introduced previously. One may question whether the first definition of the action of an element S E S ,on a basic tensor m=x,O...@x,,, x i = c a j i v jfor 1 < j < n , is consistent with the second one. According to the second definition, sm is given by

sm=x:j,c,,l . . ~ a j , c n , , , [ v j l ~ . . . ~ v j , l . However, since u j s c s - l ~ ,=~ ~ s - l ~ ithis ~ equation may be rewritten, by rearrangement of the order of the a's, as

o

I(n)[vj,o . . . vj,l sm = 1 ails- . . . ajnS= [Cajls-l(I)vjt0.'. OCaj,s-l(n)vjn] = Xs-1(1)O . . . OXs-l(,,),

which agrees with the first definition. Thus the two definitions are equivalent for basic tensors. We turn now to a connection between S,, and GL(V) where V is the rn-dimensional space introduced previously. The n-fold tensor product M of the space V with itself is a natural representation space or module for the general linear group GL(V). Each g of GL(V) is represented by the n-fold tensor product g @ . . . @g of g with itself. If vj, 0... @vj, is an element of the extended basis of M, then one recalls that (Vjl

0 .* . @ vj,)g

= vj,g 0. . .

vj*g.

The elements of GL(V) are taken to act on the right so that our results will conform to the notation introduced earlier in this section. Denote by G the group of linear transformations on M which constitute the image of GL(V) under the tensor product representation. Let D be the linear envelope of G consisting of all complex linear combinations of elements from G. Then d is a complex algebra of linear transformations acting on the space M which becomes a right d-module. Our intention is to prove that fi coincides with HomKSn(iM,M) so that the results of the first part of the section

248

4. Representation Theory of Special Groups

can be applied to this speciai case. In order to see that 0 is contained in Hom,,,?(M, M), let s E S, and g E GL(V).Then one has [ 4 v j I 0.'. Ovjn)lg =(vj,-,,,,O...Ovj,-,,,,,)g = ( y j 7 - I ( , ) g O...Ovj,-t,,)g)

= s(vjlg@ = S[(VjI @

. . . 0V j n g ) . . . 0V j , ) S ] .

Thus one finds that (sm>s = 4 m s )

(2.31)

for s E S,, , m E M, and g E GL(V). Since S, and G constitute a set of Kgenerators of KS,, and D, respectively, it follows that D is contained in HomKS,(M, MI. The fact that HomK,,(M, M) is contained in D will be demonstrated after considering a special subspace W of M. If D = HomK,,(M, M), then KS, and 0 satisfy the conditions of the previous theorems. Consequently, all the irreducible D-subspaces of M are of the form e M where e is a primitive idempotent of KS,. The most obvious primitive idempotent of KS, is the idempotent e = ( l / n ! ) s, s E S,. For any m E M, the element em has the property that t(em) = em, t E S , . since (2.31')

t(em)

=

1

t[(l/n!) sm]

=(

~ / nC ! ) tsm = em.

The subspace W of symmetric tensors consists of all w E M such that sw = w for s E S,. It follows from (2.31') that the range eM of e is contained in W. Conversely, one notes that W is contained in eM. Each basic tensor which is the product v @ . . . @ v of equal factors is clearly an element of W. More generally, if w = M(J)v(J)is any element of M, then w is an element of W iff w ( J ) = u.(sJ) for every index diagram ( J ) . This idea will be elaborated in a number of ways in later discussions so that it is worth considering in more detail. For example, let ( J ) = ( j l , j 2, j , ,j4)= (3, 1, 2, 4). Given any tensor m, m(J) is the coefficient of the basic tensor v3 @ vl @ v2 0v4 in the expansion of m. The assertion that n7(J) = m(sJ) for all s E S4, means exactly that the twenty-four basis elements v1 Ov, @ v 3 @ v 4 , . . . , v4 @ v , OV, 0v, have the same coefficient. However, if ( J ) = ( j l , j 2, j 3 ,J4)= (1, 1, 1, 2), then the implication is much less inclusive. It means only that the tensors v, Ov, @ v l @ v , , v1 Ov, @ v 2 Ov,, v, Ov, Ov, O v , , and v2 Ov, Ovl Ov, have the same coefficient. These observations suggest the introduction of an equivalence relation R on the elements of the extended basis of M. Two tensors vjl @ - - -@vj, and vk, @ . . . @ vk, of the extended basis are said to be in the relation R if and only if the sets {vj,. . . . , vjn} and {vkl,. . . , vk,} contain the same elements from

1

249

2. Modules over Symmetric Algebras

the basis {vI, . . . , v,} of V with the same multiplicity. An element m of M belongs to W if and only if its components are constant on each equivalence class of R. This point of view allows us to describe a basis and to determine the dimension of W in a natural way. First, one can select a representative element p from each equivalence class of R by choosing that element p for which the factors from {v,, . . . , v,} appear with nondecreasing subscripts. For example, the representative element from the class C consisting of v, o v , o v , O v , , v, o v 2 o v l ov,, v1 ov, o v , , v2 O v , o v , v2 ov, @ v l ov,, and v1 @ v , o v , O v , is the vector vI @ v , @ v , o v , . Second, one can assert that the dimension of W is equal to the number of classes of R which is seen to be equal to the number of unordered samples of size n taken with repetition from a population of rn elements. This number is well-known to be ( m + : - ' ) . An extension of this idea is introduced below to discuss the case of more complicated symmetry subspaces of M. We want to observe that W has a more intuitive set of generators. To do so, denote by B the set of all basic symmetric tensors of the form v 0 . . . v, v E V. The set B is a set of K-generators of the space W of symmetric tensors if every f ' E W* whirh vanishes on B also vanishes on W. Note that every linear functional f ' E W* is determined by a set { f ( J ) } of components such that if w = w(J)v(J), then

ov,,

ov,

2

f ' ( W ) = C f ( J ) w ( J )= E f ( J ) W ( S P J )

= Cf(sJ)w(J).

Consequently, one can assume without loss of generality that (2.32)

c

f(SJ)

=f(J),

S E

s,,.

Let v = Aivi denote a linear combination of the basis elements (vi} of V with the coefficients A i regarded as indeterminates. Then the basic symmetric tensor v 8.. . 8 v has the form 1Ajl . . . Ajn(vjl 0. . . 0 vj,). Let f ' be an element of W* which vanishes on B. Then (2.33)

0 =f'(v 0. . . 0v)

=Cf(ji,

. . . ,.j,,)Aj, . . . Aj,.

The terms of Eq. (2.33) can be collected into partial sums consisting of all terms with a common factor A l a l ... A,,"",CI, ... CI, = n. Two coefficients , f ( J l , , . . ,j,,) and f ( k , , . . . , k,) o f f ' multiply the same monomial A i a l . . . Anan if and only if k i = 1, 1 5 i _< n, exactly as many times as ji= 1, 1 i ~ n k i= 2 exactly as many a s j i = 2 and so forth and so on. These conditions are realized if and only if

+ +

f(ki,

. . ., k n )

= f ( j . s ( ~.).,. , j s ( n J >

that is, (2.34)

f ( W =f(sJ)=f(J>,

,

250

4. Representaion Theory of Special Groups

for some s E S , . Thus A l d l .. . An(l" appears with a coefficient C(a,, . . ., a,) equal to rf'(J) for some integer r and some index diagram J. Equation (2.33) can be rewritten (2.35)

0=

c C(Cc,,. . . ,

...

C(,)Alal

Anam,

where the 2's are indeterrninates. It follows that all of the C(CY,, . . . ,CY,)'sare zero and. consequently. all of theJ'(J)'s are zero. Thus every linear functional ,f' of W* which vanishes on B also vanishes on W. Therefore B is a set of K-generators of W. We summarize this conclusion in the form of a lemma. The set B of basic symmetric tensors of the form v 0 . .. 0v is (2.36) LEMMA. a set of K-generators of the subspace W of symmetric tensors. This result can be used to establish that the enveloping algebra b coincides with HomK,,l(M, M ) . Let Eji be the linear transformation on V whose action on the basis {,Y,. . . . , v,} is given by 1I i,,j, k I m.

(vk)Eji= a k j v i r

The set { E j , ) , I 5 i, j I m. is a K-basis of Hom,(V, V). Given two elements v I , 8.. . @ v,, and vil 8.. . 0v;, of the extended basis of M , one has (2.37)

(vk,

8. . . 0vk,)(Ej,i, 0. . . 0 Ej,, in) = (Sk,,l . . . 6," j"(vil 8.. . 0v;,,).

It follows from (2.37) in a quite straightforward manner that the set

{ E j li l 8. . . 0Ej, ,J, 1 I i, ,j , I m, is a K-basis of Hom,(M, M). We denote the tensor Ej, i , 0. ' . 0E j , by the symbol E(J, I ) and observe that

v(K)E(J. I ) = 6(K, J ) v ( I ) , where 6(K,J ) = I if ( K ) = ( J ) and otherwise is 0. We have asserted that every T E Hom,(M, M) can be written

T=

t ( J , I ) E ( J ,I ) ,

t ( J , I ) E K.

This leads to (2.38) and (2.39)

1v(s-IK)t(J, I ) E ( J , I ) =2 t(F'K, I)v(/)

[ s ( v ( K ) ) ] T =[ v ( s - I K ) ] T =

c t(K, I ) v ( I ) c t(K, I ) v ( s - l l ) c t(K, JI)V(l).

s [ v ( K ) T ]= s 1 v(K)t(J, I ) E ( J , I ) =

t(K, I ) s v ( I ) =

=s

=

25I

3. The Integral Representations of the General Linear Groups

Such an element T belongs to HomKsn(M,M) if and only if (2.40)

(sm)T = s(mT),

m

E

M , s E S,.

It follows from (2.38), (2.39), and (2.40) that T belongs to IAJmKsn(iM, I) if and only if t(s-'K, Z) = t(K, SZ)

or t ( J , I ) = t(sJ, SZ)

(2.41)

for all index diagrams I and J. Equation (2.4i) implies that T is an element of HomKSn(M,M) if and only if T i s a symmetric tensor of Hom,(M, M) regarded as the tensor product Hom,(V, v ) 8 . .. @ Hom,(V,

v),

n factors.

It follows from Lemma (2.36) that each Tbelonging to Hom,,"(M, written in the form (2.42)

T=

M) can be

c ( i ) [ t i@ . . . @ ti],

where ti E Horn#, V). The enveloping algebra is a linear subspace of HomKs,(M, M) and is therefore closed. However, every linear combination of the form (2.42) can be approximated arbitrarily near by an element (2.43)

T'

=

1 c'(i)[g,@ . . . @ g i ]

belonging to b. It follows that b is a closed dense subset of HomKSn(M,M) and therefore must coincide with it. We have the following theorem. (2.44) THEOREM. The enveloping algebra H o m K S p , M).

b coincides with the algebra

We now denote b by D to conform to the notation introduced in the earlier part of the chapter. 3. THE INTEGRAL REPRESENTATIONS OF THE GENERAL LINEAR GROUPS

This section is concerned with the application of the results of Section 2 to the determination of the integral representations of the full linear group GL(V) on a complex m-dimensional space V. An integral representation T of GL(V) is a matrix representation in which the entries of the representing matrix T(g), g E GL(V), are polynomials in the elements of the matrix of g with respect to any basis of V. All integral representations of GL(V) are

252

4. Representation Theory of’Special Groups

completely reducible. They decompose into irreducible components equivalent t o the canonical irreducible CL(V)-submodules N(F, V) obtained by reducing the tensor representations of GL(V) on the tensor product M = V @ . . . @ V. Such irreducible modules are generated by the primitive idempotents of KS,, acting on M. The primitive idempotent ( ] / t i ! ) s, s E S , generates the irreducible GL(V)-submodule W of symmetric tensors. Usually. it is sufficient to determine the primitive idempotents up to a nonzero scale factor, so that we deal mostly with essential idempotents to avoid unnecessary factors. However, one must insert the scale factor in special instances. The analysis parallels that for the symmetric subspace W, but requires an elaborate notational scheme yet t o be established. Given a primitive idempotent e E K S , , an element m belongs to the irreducible right D-submodule eM if and only if (3.1)

em

= m.

Since the analysis of Eq. (3.1) is complicated, we introduce the basic ideas for a special case. Let M denote the sixfold tensor product of an rn-dimensional vector space V with itself. The irreducible right D-submodules of M are determined by the primitive right ideals of s6. Given a frame associated with the partition 3 + 2 + 1 = 6, consider the Young tableau

123 T=45. 6 The group P(T) of row transformations of this tableau is generated by the set {(12), (13), (23), (45)) of transpositions and the group Q(T) of column transformations by the set {(14), (16), (46), (25)). It is unnecessary for our purposes t o write out the full expansion of the essential idempotent e(T) = &(y)qp,sometimes denoted merely by e. An element m E M has the general form

C

(34

C n?(jl,. . .,j 6 ) ( v j ,o . . . @ vj6).

This equation is frequently replaced by one in which the Young’s tableau defining e is more explicitly displayed. Thus one uses (3.2‘)

0 vj2 0

3. The Integral Representation of the General Linear Groups

253

The symbols (3.3) are referred to as index diagrams. Again we find it useful to denote such diagrams by a single symbol as (3). An element s E S, acts on h e subscripts of such a diagram as (3.3) to produce a new diagram (3.3') These diagrams clarify the relationships between the components of an element m of M and those of sm, s an element of s6. One has jl j2 J3

A tensor m is said to be symmetric in its rows, with respect to the tableau 123 45 6

5

if and only if m

=p m ,

that is,

for every p belonging to the group P(T) of row transformations of 123 45 6 I

A tensor m is antisymmetric in its columns if and only if &(q)m= qm, that is,

254

4. Representation Theory of Special Groups

for every q belonging to the group Q(T) of column transformations of the tableau 123 T=45

6

ithiclz is taken to heJixetl in the following discussion of our special case.

The usual treatment, according t o Boerner and others, of the subspace U = eM determined by Eq. (3.1) is through an intermediate subspace W = PM of the tensor product space M. We retain the symbol W for this space since W is a symmetric space in the following sense. Let w = Pm. Note that p w = pPm = P m = w, p E P ( T ) , so that w is symmetric (invariant) under any permutation p from the row group P(T). One sometimes says that W is symmetric under the r 0 ~1 -of s the given tableau. Every element u E U = QPM is the image Qw of an element w of W. The equation u = Qw can be used t o determine a set of equations for the components of the image u in terms of those of w, namely, one has

u

= = =

C ~01,. . . ,j6)(vj10 * . .@ v j 6 )= QW Q 1 Lc(j1, . .. v.j6)(vj, O

(C & ) i i . ( j y ( l )

9

Ovj,) .. jq(6)))(yj, O - * '

' '

. 7

. O vj6).

These lead t o the component equations: (3.6)

4 j l , . . . ,id = C & ) 4 j y ( l ) . . .

jq(6)h

where q runs through the elements of the column group Q(T). Since the components of the elements of W = P M are symmetric in their rows, one has (3.7)

.il

j, j3

for p E P ( T ) . In contrast, the components of the elements of U = QW are antisymmetric in their columns. If u = Qw, then gu = qQw = E(q)Qw = E ( ~ ) uso , that (3.8)

jl j 2 j 3 4 q h j4 j 5

li6

I lJq,6)

jy(l)

jq(2)

= u 1q(4)j q ( 5 )

jq(3)

I

for q E Q(T). The symmetry of the components of the elements of W in their rows and the antisymmetry of the components of elements of U in their columns play a key role in our considerations.

3. The Integral Representations of the General Linear Groups

255

According t o Eq. (3.6), the components of u belonging to U = QPM are linear forms in the components of tensors w lying in W = P M . Our problem is to determine a basic set of these forms. We sort out the components of an element w of W into equivalence classes. An index diagram J is equivalent t o an index diagram J’ if and only if J ’ = pJ for some p E P(T). This is easily seen to be an equivalence relation A partitioning the set of all index diagrams associated with the components of tensors of W into equivalence classes C. One notes that two index diagrams J and J‘ are equivalent if and only if each row of J contains the same integers with the same multiplicities as the corresponding row of J’. For example,

112 J=23 4

and

121 J1 = 32 4

are equivalent index diagrams which are not equivalent to

122 J2=23 . 4 The index diagram J2contains the integer 2 twice in the first row while the index diagrams J and J1 contain it only once. Observe that a tensor m of M is a n element of W if and only if the components of m are constant on the equivalence classes of the relation A. This follows since Eq. (3.7) shows that if w E W, then w(J) = w(pJ). Conversely, if m(J) = m(pJ) for all p E P ( T ) and index diagrams J, then p m = m, which implies that m = ( l / I P ) ) P mis an element of W, where ]PI denotes the order of P(T). One selects a representative diagram J R from each equivalence class C of A according t o the rule that the entries do not decrease in any row of J R . For example,

112 JR= 12 3 is the representative element of the class

121 121 211 112 112 211 21 , 12 , 21 , 12 , 21 , 12 3 3 3 3 3 3 The representative element from each equivalence class is called a rowordered index diagram. The components corresponding t o the row-ordered index diagrams can be taken as the independent components determining

256

4. Representation Theory of Special Groups

the subspace W = P M . The equations defining this subspace by means of its components assert that each component with index diagram belonging to the class C is equal t o the component with row-ordered index diagram from that class. The determination of the independent components of tensors belonging to the space U = QW is a good deal more complicated, although the final result is equally easy to describe. The independent components of tensors in U can be chosen to be those with index diagrams in which the entries d o not decrease from left to right along the rows and increase as one proceeds down each column. Such index diagrams are called standard index diagrams. The discussion proceeds in three steps. First, one shows that components with standard index diagrams in which all the indices are distinct are linearly independent. Then one shows that components with nonstandard diagrams containing these same indices depend linearly on those components with standard index diagrams. Finally, one considers the case in which the indices are not necessarily distinct. Assuming that m 2 6. it is no special restriction to consider first the case in which the set { j , . . . . ,j o ] consists of the integers 1 through 6 so that the standard index diagrams have the indices

123 123 124 124 125 125 126 126 45 46 35 36 34 36 34 35 6 5 6 5 6 4 5 4 and

134 134 135 135 136 136 145 146 25 26 24 26 24 25 26 25 , 6 5 6 4 5 4 3 3 each of which corresponds t o one of the standard tableaux for this frame. These are listed as an increasing sequence with respect to their usual ordering. Each component u ( 0 , ) corresponding t o a standard index diagram D,, 1 5 i 5 16, can be expressed as

(3.9)

u(Di) =

1& ( 4 ) ~ ( 4 D i ) .

Since the effect of any q E Q(T) is t o permute the integers of D i , those integers which appear in 4 0 , are exactly those belonging t o D,.One must keep in mind that 4 acts on the subscripts of the entries of D i , that is, (3.10)

jL

j 2

j 3

25 7

3. The Integral Representations of the General Linear Groups

Note, moreover, if q

= (146)

is preceded b y p

= (123),

then

so that q does not act as a column transformation on the index diagram at the second step. The elements from P(7‘) and Q(T) act as YO,, and column transformations on the basic tensors and not necessarily as row and column transformations on the index diagrams. Given a basic tensor of the form

the action of p E P(T) is always t o permute elements which belong t o fixed rows in the basic tensor and that of q E Q(7’) is to permute elements which belong to fixed columns. In the case at hand, where p = (123) and q = (146),

If

a

@

b

(4P) d 0 e

@

c

I If

c @ a @ b

=qd

@

e

However, when one considers the action of q p on an index diagram E, the action of q, as we have seen, may not be t o permute the indices which belong t o a fixed column ofpE. To return to an example of (3.9), consider u ( D 1 ) where

123

D,= 45 . 6

258

4. Representation Theory of Special Groups

Since the ii>(Ei)are symmetric in their rows, each component occurring on the right-hand side of (3.11) with an index diagram E i which is not rowordered can be replaced by the corresponding row-ordered component without altering the value of the component iv(Ej). The right-hand side of (3.11) appears with the row-ordered index diagrams

123 123 234 234 236 236 45 56 15 56 15 45 6 4 6 1 4 1 and

135 135 345 345 356 356 24 26 12 26 12 24 6 4 6 1 4 1

1 1 I 1( 1

after these substitutions. Thus one finds that 141

I23 ;t5

= it1

123 135 ;t5 - it.1 f

135

+

+ R1,

where R , denotes the sum of those components w(EJ with E , a row-ordered but not a standard index diagram. The next step is t o show that the components u ( D l ) , . . , , u(D,,) are linearly independent. The process of row-ordering the index diagrams according t o this scheme is repeated for the remaining standard components of u to obtain sixteen equations:

+ ... + U I , I ~ ~ ~ (+DR,I ~ )

u(D1) = a , , ~ ( D l )

+

u(D2) = u ~ I I v ( D I ) . . .

(3.12) u

~

D

l

~

~

~

a

l

~

,

+ l

+ Rz

a2,16 l ~ ( D 1 6 )

~

~

~

~

D

~

~ R~1 5 ~

~

'

~

a

l

~

z4(D16)=a1~,~rc'(D1)+' " + a 1 6 , 1 6 ~ 1 ' ( D ~ 6 ) + R ~ 6 ~

Consider the system of Equations (3.12) as defining sixteen linear forms,

u ( D , ) , . . . , u ( D I 6 ) ,in the linearly independent, row-ordered components of tensors of W. These sixteen forms are linearly independent if the matrix A = {trijj, I I i, j I 16, has a nonzero determinant. One demonstrates this by showing that A is an upper-triangular matrix with all 1's along the main

diagonal. (3.13) REMARK.Each element a i i . 1 < is 16, arises from the action of q = 1. e(q) = 1 , since, as we shall see, only the identity permutation followed by no alteration of the rows leaves any D iunchanged. Thus one has a,, = 1,

,

1

6

M

'

259

3. The Integral Representations of the General Linear Groups

1 5 i 5 16, as stated. A component w(E)occurs on the right-hand side of the equation defining u(Di) if and only if the index diagram E arises by rowordering q D i for some q E Q(T). For example, let q = (146) E Q(T) act on

il j4

J2

h

j3

.i,

to give

j4

h

j, j,

j3

jl

In particular, consider 125 D , = 36 ; 4

then

325 qD6 = 46 . 1

The row-ordered diagram E eventually obtained from D , is

235 46 1 which results from interchanging the indices j , = 3 and j z = 2 by means of the permutation p’ = (24). Note that p’ is not a row transformation on

123 45 . 6 However, one can write the result as p‘qD, = q(q-’p’q)D6 p = (164)(24)(146) = (12) is a row transformation of

= qpD,

, where

123 45 . 6 More generally, the row-ordered index diagrams which occur on the righthand side of each equation of u(Di) in the system (3.12) arise from D i by the application of a qp, p E P(T),q E Q(T), where P ( T ) and Q(T) denote the row and column groups on the Young’s tableau

123 45 . 6 No row-ordered diagram Ei appears on the right-hand side of the equation for u(Di) more than once. Otherwise, one has qpDi = @PDi for distinct pairs (q,p ) and (ij, p”). We have seen in Section 1 that this is impossible. Thus the a i j are all equal to 1, 0, or - 1.

260

4. Representation Theory of Special Groups

(3.14) REMARK.Suppose now that p ' q D i = D k , where D, is a standard index diagram different from D i. We assert that D i < Dk in the usual ordering of the diagrams. To see this, denote by r the first row (counting down from the top) of D ialtered by the permutation q. An element of r changed by q is replaced by a larger element from below. After row-ordering q D i t o obtain D,, the first element of the new row r' corresponding to r must be at least as large as the first element of r , the second element of r' must be at least as large as the second element of r , and so on. Since r' differs from r, one must eventually reach an element of r' which is larger than the corresponding element of Y. This proves that D i< L f , . These observations establish our claim that A is upper-triangular with 1's on the main diagonal. Thus det A # 0 and the forms u(D,), . . . , u( Dl,) are linearly independent. There remains the task of demonstrating that the linear form u(E), E a nonstandard diagram whose indices make up the set {I, . . . , 6}, is a linear combination of the forms u(D,), . . . , u(D,,). This argument depends on of KS,, which was introduced by H. Weyl. Make correspond an involution to each x = x(s)s, s E S, , of KS,, an element I defined by

-

1

(3.15)

2

-

=

c x(s)s-I.

G=

I t is not difficult to see that is an involution on KS , such that YI. In particular, E is a primitive idempotent if and only if e is a primitive idempotent. To be specific, the primitive idempotent e = c(q)qp corresponds to the primitive idempotent E = c(q)p-'q-' = &(q)pq. The second equality holds since P(T) and Q ( T ) are closed under inverses and ~ ( q=) e(q-'). The right ideal eA defining the subspace U = eM corresponds t o the left ideal AE under this involution. In particular, eA and A&have the same dimension over K. There is a close relation between AE and eM discovered by Weyl. For any u E U, let C(E)E KS,, be defined by

2

E(E) =

(3.16)

1 2

1 u(sE)s

for some fixed index diagram E. The element E(E) of KS, is called a ring tensor component. (3.17) THEOREM. The set of all ring tensor components arising from components of tensors u contained in the irreducible right D-submodule U = eM, e denoting a true idempotent @)/A, belongs to the minimal left ideal AE generated by the involute E of e. Proof. An element u of M belongs to U if and only if eu = u, that is, if and only if seu = su for every s E S,, . Hence for any diagram E (su)(E)= ( s 4 ( E ) =

c

& ( q ) ( s q p w ) l A=

c 4q)u(sqpE)/A.

3. The Integral Representations of the General Linear Groups

Also

u"(E)&= = =

c u(sE)s1&(q)pq/I. cc

=

26 I

1 2 &(q)u(sE)spq/i,

&(4)u(t4-'P-1E)tlA =

c c 4q)+?pE)r/A

2 (tll)(E)t= 1 u(tE)t = G(E).

Thus C(E) E (KS,)C for every u E U and every index diagram E. This result allows us to prove that components u(E) determined by nonstandard index diagrams depend linearly on the standard components. Let E be a nonstandard index diagram with indices the set { 1, . . . , 6). The ring tensor component G(E) = u(sE)s is an element of A&which is sixteendimensional. There exists a linear relation among the components u ( E ) , u(D,), . . . , "(D,,). Since fu(D,), . . . , u(D,,)} is a linearly independent set, this relation can be expressed in the form

c

(3.18)

u(E) = a,u(D,)

+ . . . + a,,u(D,,).

The coefficients in Eq. (3.18) can be evaluated by elimination between Eqs. (3.6) and (3.12) which we repeat for convenience: (3.6)

u(E)

=

c &)w(qE)

and

(3.12)

t i ( D i )=

aijw(D,)

+ Ri,

Note that Eqs. (3.12) can be solved for (3.19)

1

~ ~ (= o j )bj,u(D,)

it,(Dl),

+ Kj,

1I i I 16.

. . . , it(D,,) to obtain 1 ij I 16.

Here B = {b,J, I I j , s I 16, is an upper-triangular matrix and K, denotes the sum of the iv(F) with F row-ordered, but not a standard index diagram. It is sufficient t o consider Eq. (3.6) for the case of components with columnordered index diagrams E since the components of a tensor u in the space U are antisymmetric in their columns. Then one notes that when E is a columnordered index diagram, the process of row-ordering can be carried out without disturbing the fact that the final diagram is column-ordered. To see this, let E be a column-ordered index diagram with m rows, all of \c,hose indices a r e distinct. Clearly, one can row-order the mth row by interchanging whole columns to obtain a new index diagram Em which is column-ordered and whose rnth row is row-ordered such that w(E,,,) = w(E). Suppose one has reached an index diagram E,,, which is column-ordered with rows r + 1 through m row-ordered and IV(E,+~) = ~ i t ( E )If . there exists i < k such that row r contains b < a with a in column i and b in column k , then one has a

262

4. Representation Theory of Special Groups

situation similar t o the index diagram E,,, sketched as follows: i

k

XI

Yl

X,-I

Y,-I

a c

with the significant columns

h rowr d rowr'l

with b < (1 < c < ti. We permute E,,, t o a new index diagram E,' by means of a rowz permutation 17' interchanging the first I' elements of column i with the corresponding first Y elements of column k . This gives a diagram with the columns: i

k

which is still column-ordered since h < c and a < (f. After a finite number of such permutations, one arrives at an index diagram E, which is not only column-ordered. but also row-ordered in rows r through m and for which it.(E) = ii.(Er).By repeating this process, one arrives at a standard index diagram F with iifE) = i i . ( F ) . Since ir.(E) = it(qE). q = I . it follows that the equation u ( E ) = E(q)it.(qE) contains components corresponding to standard index diagrams on the right. Thus one can rewrite the above equations as u ( E )=

(3.20)

c j \i.(Dj) + K ,

where K denotes the sum of those components of 14' belonging t o row-ordered, but not standard index diagrams. Replacing each component it.( D j ) of (3.20) by its value from (3.19), one obtains (3.21)

u(E)= =

1 c j [ C hj,u(D,) + K j ] + K [Icjhjs]u(D,)+ 2 c j + K. Kj

I

When we compare Eq. (3.18) and (3.21), we discover that cj K j + K = 0. Otherwise. there exists a nontrivial linear combination of u ( D , ) , . . . , u(D,,) which is linear in the nonstandard, row-ordered components of w. This con-

3. The Integral Representations of the General Linear Groups

263

tradicts the fact that the coefficient matrix {ajj} of Eq. (3.12) is nonsingular. Thus one finds (3.22)

u(E) =

1 1 cjbjsu(Ds)>

where the {cj} are determined by Eq. (3.20) and the {bjs}by Eqs. (3.12) and (3.19). The actual determination of these coefficients is straightforward for small values of rn and n, but we defer specific examples for the moment. See Examples (3.43) and (3.44) below. We have completed two-thirds of the program for our special case, namely, that the forms u ( D j ) , 1 I iI 16, are linearly independent and that any form u(E) where E is a diagram whose indices are the set { I , 2, 3. 4, 5, 6) can be expressed linearly in terms of these. The forms u(E) where E contains distinct indices, not necessarily belonging to the set { 1,2, 3, 4, 5, 6}, can be treated in a similar manner. Such a u(E) is the coefficient of vj, vj,

0 vj2 0 vj3 0 vj3

'j,

where the component vectors {vji}, 1 5 i 6 , are all distinct. By renaming the basic vectors of V, such components u(E) coincide with those just discussed. There remains the question of components u(E) with repetition among the indices of the index diagrams E. To fix the idea, consider diagrams for which the indices are 1, I , 2. 2, 2, 3. These diagrams determine coefficients which are multipliers of basis tensors of the general form v1 v, v3

0 v1 0 v2 0 v2

including all basic tensors obtainable from it by permutation of its factors. There exists a linear transformation T on V such that Tv, = v l , Tv, = vl, Tv, = v 2 , Tv, = v 2 , Tv, = v 2 . and Tv6 = v 3 . The linear transformation T 0. . . 0T, six factors, converts any linear relation among the basic tensor v1 v4

0 v, 0 v3 0 v5

'6

and its permutations into a linear relation among the basic tensors v1 v2 v3

0 v1 0 0 v2

v2

and its permutations. It follows that Eqs. (3.12). (3.19), (3.20), and (3.22)

264

4. Representation Theory of Special Groups

remain valid for such index diagrams. Some equations among this set become redundant and some trivial, but they remain valid. The discussion which we have given of our special case extends with little change to the general situation. We must refer the reader t o more specialized treatises for the details. I n summary, the action of elements of S, on elements of M = V 0 . . . 0V, n factors, has been defined. Each canonical tableau T defines a n essential idempotent e = e(T) = QP. This essential idempotent P(T) determines an irreducible GL(V)-submodule of M according t o Eq. (3.1). The expansion of Eq. (3.2’) holds with the components m(J) of m designated by means of index diagrams ( J ) corresponding t o the frame F of T. The concepts of symmetric in tlir rows and antisymmetric in the colt.lmns remain valid with the obvious modifications. The space W of “symmetric” tensors is introduced, and Eq. (3.6) assumes the form (3.23)

44 =

1 4q)l4qJ),

for the components of a tensor u E U analog of Eq. (3.12), namely. (3.24)

=

4

QPM

+

u(Di) = ~ a j j w ( D j )R , ,

QV), =

QW, where u = Qw. An

1 I i
holds for the components designated by standard index diagrams{ D,}, 1 5 i I is the dimension of the minimal right ideal e(T)A. Again, one finds in general that these components constitute linearly independent forms with variables in the independent components of the symmetric tensors of W. Furthermore, they turn out t o constitute a basis for components with index diagrams whose indices are the set { 1, . . . , n}. Results analogous t o those of our special case hold for components u(E) when E has distinct indices not necessarily belonging t o the set { 1, . . . , n) and also for cases where E contains repeated indices. We formulate, without proofs. some of the principal results as theorems. Let M denote the GL(V)-module V 0.. . 0V, n-factors, where V has dimension 111. Let {m,, . . . . n7,] be a partition of n determining the frame F with k 2 m, and let T be the canonical tableau belonging t o F. Then e(T)M = N(F, V), c ( T ) the essential idempotent determined by T, is a n irreducible GL(V)-module. One defines a new sequence {ml’, . . . , n?,’} whenever k < m by taking mi’ = 1 1 1 , . 1 I i c; k. and mi‘ = 0, k < i I n7. Let the set {A,] be defined by

A corresponding t o standard tableaux. Heref

(3.25)

ibi= mi’+ m

-

i,

I I i 5 m.

(3.26) DEFINITION. The GL(V)-module N(F, V) is called the canonical GL(V)-177odule dcrerrnined hy F and the representation T(F, V) it affords the canonical representation.

265

3. The Integral Representutions of the General Linear Groups

THEOREM. The dimension d of the canonical GL(V)-module N(F, V) is equal t o the number of standard index diagrams which can be put into the frame F.

(3.27)

THEOREM. The dimension d of the canonical GL(V)-module N(F, V) is given by

(3.28)

d=

A@,, . . . ,%,) A(m-1, ..., 1,O)'

where Ai, 1 5 i I m, is given by Eq. (3.25) and A(zl, . . . , z,) is the difference product of the integers z1 through z, . The difference product A(z,, . . . , z,) is defined to be ( ~ 1-

z2)(z1 - ~ (22

(3.29)

... (

~ 1- Zm)

3 )

- z3) . . . (z2 - z,)

(zm - 1 - zm>. (3.30) EXAMPLE. Let V be a four-dimensional vector space and M be the three-fold tensor product V @ V 0 V. There are three partitions of 3, namely, {3}, (2, l}, and (1, 1, 1) to which correspond the frames Fl = '.. , F2 =

..

. , and F3 =

: . We consider

each of these frames in turn.

Let TI = 123 be the canonical tableau of Fl

=

R l . The stan-

dard index diagrams of F,, in ascending order, are 111

112 113

222 223

114 122 123

224 233

124

234 244 333

133

134 144

334 344 444,

which shows, by Theorem (3.27), that the dimension of N ( F , . V) is 20. Now, one has nz,' = 3, m2' = 0, m3' = 0, and m4' = 0, so that

+ 4 - 1 = 6, A2 = 0 + 4 - 2 = 2, % =0 ,+ 4 - 3 = 1, A4 = 0 + 4 - 4 = 0. EL, = 3

Consequently, d = A(6, 2, 1,0)/6(3,2, 1,O)

=

266

4. Representation Theory of Special Groups

Thus the results of Theorem (3.28) agree with those o f Theorem (3.27). (b)

Let T ,

=

l 2 be the canonical tableau of F2 = 3

. The stanU

dard index diagrams of F 2 , listed in ascending order, are

11 1 1 1 1 12 12 12 13 13 13 14 2 3 4 2 3 4 2 3 4 2 14 14 22 22 23 23 24 24 33 34 3 4 3 4 3 4 3 4 4 4

so that the dimension of N(F,, V) = 20. I n confirmation, ,Il = 5, ,Iz = 3, 2, = 1 , and A4 = 0, so that, by Theorem (3.28), d = A(5, 3, l,O)/A(3, 2, 1,0)= ( 5 - 3)(5 - 1)(5 - 0)(3 - 1)(3 - 0)(1 - 0) (3 - 2)(3 - 1)(3 - 0)(2 - 1)(2 - 0)(1 - 0)

(c)

Let T ,

1

=2

3

be the canonical tableau of F, =

= 20.

U. The

H

standard

U

index diagrams of F, are

1 1 1 2 2 2 3 3 3 4 4 4 In this case, one has 1, = 4, ,Iz = 3, 2,

= 2,

and

A4

= 0.

This gives

d = A(4, 3, 2, O)/A(3, 2, 1, 0) = 4. The canonical GL(V)-modules play a fundamental role in the integral representation theory of the general linear group. For fixed V, GL(V), and M, the isomorphism class of the GL(V)-module N ( F , V) depends only on F. Given a tableau T‘ different from the canonical tableau T of F, the GL(V)module r(T’)M will generally be distinct from the GL(V)-module c(T)M. However. there exists s E S, such that T‘ = sT, so that e(T’) = se(T)s-l. This implies that Ae(T’)s = Ae(T). which shows that Ae(7’) and Ac(T) are A-isomorphic right ideals. Consequently. e(T)M and e(T’)M are GL(V)isomorphic right GL(V)-modules. As a result, there are at most CI distinct classes of isomorphic, irreducible canonical GL(V)-modules where x is the number of partitions of n. the number of times that V occurs as a factor of $1. Tlirrtj are less than this nutnbrr when V has dimension m less than n.

3. The Integral Representations

of

26 7

the General Linear Groups

(3.3 1) THEOREM. Every integral representation of the general linear group GL(V) is completely reducible and decomposes into the direct sum of irreducible GL(V)-modules, each of which is isomorphic to one of the various canonical GL(V)-modules N ( F , V) determined by different choices of the frame F. (3.32) REMARK. There are other classes of representations of the general linear group GL(V) for which the theorem of complete reducibility fails. The reader may have noticed a crucial omission in our previous remarks. Suppose that V is a vector space of low dimension. For instance, let V have dimension one. Then M has dimension one for every n 2 1 and can serve as a representation space only of the identity representation or the alternating representation of S,. Note that the alternating representation of S, is a one-dimensional representation in which the even elements of S, map into the identity and the odd elements into its negative. Thus one suspects that for such low dimensional V, most of the idempotents e(T), T belonging to an integer n greater than 1, act as the zero transformation on M and produce only the trivial GL(V)-module (0). The precise theorem is as follows. (3.33) THEOREM. A canonical tableau T belonging to the frame F determines a nontrivial canonical GL(V)-submodule N(F, V) of M = V 0 . . . 0V, n factors, if and only if the number of rows of F does not exceed the dimension in of V. There is another interesting duality between the decompositions of M = V 0. . . 0V as a left KS,,-module and its decompositions as a right GL(V)module. By complete reducibility, one can write (3.34)

M = L, @ . . . OL,,

where each L, , 1 5 i 5 k , is an irreducible left S,-module, or as (3.35)

M = R, @ . . . OR,,

where each R i , 1 I i I s, is an irreducible right GL(V)-module. One can reorder the elements of (3.34) so that the first 2 members contain exactly one element from each of the distinct classes of isomorphic S,-modules represented in this decomposition. Let N i be the direct sum of the n , summands of (3.34) which are isomorphic t o the mi-dimensional So-module L , 1 2 i i i. Thus one finds that ~

(3.36)

M

=

N, @ . . . O N , ,

each N i , 1 I iI 2, the direct sum of n, irreducible summands, all mutually S,-isomorphic and of common dimension m i .In a similar fashion, one finds that M = P, @ . . . @P,, (3.37)

268

4. Representation Theory of Special Groups

each P i , 1 5 i 5 rn, the direct sum of r , irreducible summands, all mutually GL(V)-isomorphic and of common dimension t i The irreducible GL(V)submodules which are the direct summands of Pi are not isomorphic to the direct summands of Pj , i # j . (3.38) THEOREM. Each N i , 1 I i 5 %,of (3.36) is a right GL(V)-submodule of M. Furthermore, N, is the direct sum of m imutually isomorphic irreducible right GL(V)-submodules, each of which is of dimension n i ,1 5 i 5 A. The duality, of course, is in the interchange of number and dimension in passing from irreducible S,-submodules to irreducible GL(V)-submodules. There is a companion theorem, which follows. (3.39) THEOREM. Each P i , 1 < i < m, of (3.37) is a left S,-module of M. Furthermore, P i is the direct sum of t i mutually isomorphic left &-submodules, each of which is of dimension r i , 1 < i < m. (3.40) REMARK. Given M = V 0. . . @ V (n factors), the canonical tableau T of a frame F belonging t o n determines a nontrivial canonical GL(V)module N(F, V) if and only if the number of rows of F does not exceed the dimension rn of V. One can compute the number of standard tableaux which belong to the frame F thereby determining the dimension f of the minimal right ideal e(T)A. The number f is the number of times the canonical right module N ( F , V) occurs in M. Conversely, one can determine the dimension d of N ( F , V) by Theorems (3.27) or (3.28). The number d is the number of times the irreducible left S,-module AZ(T) occurs in M. The module N ( F , V) occurs in M if and only if e(T)M # {0}, where T is the canonical tableau corresponding to F. (3.41) EXAMPLE. The previous theorems and remarks can be applied to Example (3.30). The frames in question are Fl = ... , F2 = , and F3 = . The standard tableau corresponding to Fl is 123; the standard tableaux ’

corresponding to F2 are l 2 and 3 1

1’;

and the standard tableau corresponding

t o F3 is 2. Thus the corresponding minimal right ideals e(T,)A, e(T,)A, and 3 e(TJ.4 have dimensions one, two, and one, respectively. These dimensions reveal that N ( F l , V) occurs in M once, N ( F 2 , V) occurs in M twice, and N ( F , , V) occurs in M once. Thus (3.42)

M

= N(F1, V)

0 2N(FZ1 V) 0N ( F 3 , V),

3. The Integral Representations of the General Linear Groups

269

so that the various dimensions are seen t o balance. Conversely, one sees that AZ(Tl) occurs twenty times in M, AZ(T,) occurs twenty times in M, and AZ(T,) occurs four times. Again, the dimensions check. (3.43) EXAMPLE. Consider the three-fold tensor product M = V 0V 0V, where V is a two-dimensional vector space. Then M is an eight-dimensional vector space affording representations of both S , and GL(V). The partitions of 3 are {3}, (2, l), and (1, 1, l}, which determine the frames Fl = . . . , F2 = '

, and F3 = : . The canonical tableau Tl = 123 of Fl has the row group

P(Tl) = ((l), (12), (13), (23), (123), (132)) and the column group Q(Tl)= (( 1)). The essential idempotent of Tl is e(Tl) = (1) + (12) + (13) + (23) + (123) + (132).

The invariant subspace W = c(Tl)M is the space of symmetric tensors. This is typical. The frame corresponding to the trivial partition {n} of n > 1 always defines an idempotent determining the subspace W of symmetric tensors of M . The dimension of W, by an earlier observation, is ("+:-') which gives ($) = 4, in our particular example. For such a simple case, one can determine a basis of W by observation. It consists of

0v1 0 v1, w2 = v1 0v1 0v2 + v1 0v2 0v1 + v2 0v1 0vl, w3 = v1 0 v2 0 vz + v2 0v1 0 v2 + v2 0vz 0v1. w4 = v2 0 v2 0v 2 . It is easy to see that Wl = (wJ, W, = (w2), W, = (w3),and W, w1

= v1

= (w4)are one-dimensional subspaces of W, each of which affords the identity representation of S,. Since the column group Q(T)= {(l)}, it follows that U = QW = W in this particular instance. Our previous remarks on duality imply that U affords a four-dimensional, irreducible representation of GL(V). The subspace U can also be described by standard index diagrams which determine a full set of linearly independent components of any u E U. These standard index diagrams are J1 = (1, 1, I), J , = (1, 1,2), J , = ( I , 2, 2), and J4 = (2,2,2). A general element u of U is determined by the component equations:

u(1,2, 1) = u(l, 1,2), 4 2 , 1, 1) = 4 , 1,2), 4 2 , 1,2) = 4 1 , 2 , 2 ) , @,2, 1) = u(l,2,2), where the components u(1, I , I), u(1, 1,2), u(1, 2,2), and u(2,2,2) are arbitrary.

270

4. Representation Theory of Special Groups

- l 2 of F2 = : , has a row groupP(T,) 2-3 column group Q(T2)= {(l), (13)). Thus

The tableau T

'

=

= {(l),

(12)) and a

QP = [(I) - (13)1[(1) + (1211

which expands to ( I ) + (12) - (13) - (123). The frame : corresponds to the partition (2, 1 ) so that, by Eq. (3.25), one has rn, = 2, rn, = 1, 2, = 2 + 2 - 1 = 3, and i., = 1 + 2 - 2 = 1. Thus the dimension of the canonical module N(F2 , V) is given by d = A(3, l)/A( I , 0) = 2, according to Theorem '

(3.28). The standard index diagrams of F2 are J

- I and J2 = :2. The '-2 only nonzero components of elements u of QPM are u(:'), u(:'), u(:'), and u(:'). The component equations are

21 4, 1 = -u(:'>,

( u y ) = -u(:')

with 4;') and u(:') arbitrary. Since the dimension of N ( F 2 ,V) is two, it follows that the dual representation of S , occurs twice in M. The dual representation is also two-dimensional so that N(F, , V) is of multiplicity two as an irreducible right GL(V)-submodule of M. Thus

M

N(F i V) 02N(F2 > V),

1

3

so that F , determines only the zero GL(V)-submodule of M . This result checks with Theorem (3.33). In confirmation, one notes that the components of tensors transforming according to F, must be simultaneously skewsymmetric in all indices and have a repeated index-thus must be zero. The foregoing example does not employ our machinery in much depth, so that we introduce an additional example. (3.44) EXAMPLE. Consider the four-fold tensor product M of a four-dimensional vector space V with itself. This space M has dimension 256. The number 4 has the five partitions: (41, (3, l}, ( 2 , 2) (2, I , l}, and { I , 1, 1, I}, listed in descending order. These determine five frames:

Fl

=

.... ,

F2

- .

" '

F , --. . ,

"

7

F 4 = : ,

and

F,=:.

According t o Theorem (3.33), each of these determines a canonical, irreducible GL(V)-submodule N ( F , , V), 1 i 5 5. of M. Using Theorem (3.28), one finds that dim N(F,. V) = 35, dim N ( F 2 , V) = 45, dim N ( F 3 , V) = 20, dim N(F,, V) = 15, and dim N ( F , , V) = I . According to Theorem (2.31'), the dimensions of the irreducible representations of the symmetric group S , belonging to the frames F,, F 2 , F 3 , F 4 , and F, are the number of stan-

3. The Integral Representations of the General Linear Groups

271

dard tableaux, 1, 3, 2, 3, I , respectively, associated with them. This means that

by our duality theorems. We examine the canonical, irreducible GL(V)-submodule N ( F 2 . V) in more detail, The partition (3, 1) gives the sequence (3, 1,0,0) which determines I , = 6, I , = 3, I , = I , and I., = 0. Thus one has dim N ( F 2 , V) is d = A(6, 3, I , O)/A(3, 2, 1, 0) = 45. We verify this number by writing down the full list, in ascending order, of standard index diagrams for the frame F 2 . 111 111 I l l 2 3 4

112 112 112 113 113 113 2 3 4 2 3 4

114 114 114 122 122 122 123 123 123 2 3 4 2 3 4 2 3 4 (3.45)

124 124 124 133 133 133 134 134 134 2 3 4 2 3 4 2 3 4 144 144 144 222 222 223 223 224 224 2 3 4 3 4 3 4 3 4 233 233 234 234 244 244 333 334 344 3 4 3 4 3 4 4 4 4

The canonical tableau T2 = T = 123 of the frame F2 has for its column 4 group Q(T) = {(l), (14)) which makes the applications of Eqs. (3.6). (3.12), (3.18), (3.19), and (3.21) rather simple. There are five classes of index diagrams associated with F, , namely, (a) all indices the same. (b) three the same and one different, (c) two of one value and two of another, (d) two of one value 111 and two which are distinct, and (e) all distinct. Examples of these are 1 11' '12 123, respectively. Since elements u belonging to U = QW 2 2 3 4 have components skew-symmetric in the columns of the index diagrams, it follows that components u(jii) of type (a) are always zero. Components of type (b) are zero for the same reason, except in the case of u(;ji) or u(j"), where i # j . Thus the last three types are the ones of most interest. We recall that Eq. (3.6) relates components of u = Qw, w E P M , only when their index diagrams contain exactly the same indices with the same multiplicities

272

4. Representation Theory of Special Groups

although, of course, not in the same arrangement. This means, for instance, that the components of u with index diagrams from (3.46)

112 121 122 211 212 221 2 2 1 2 1 1

are linearly related, but they are not related to components of u whose index diagrams have indices belonging to a different set. Since there is only one standard index diagram in (3.46), each component of u with an index diagram from (3.46) is a multiple of u(i”). Let an element u of QW be of the form Qw. Then one has, by Eq. (3.6), u ( J ) = H ~ J-) ~ ( ( 1 4 ) J ) for every index diagram J . In particular,

A typical example of type (d) is given by the set of index diagrams

(3.47)

112 113 121 123 131 132 3 2 3 1 2 1 211 213 231 311 312 321 3 1 1 2 1 1

The standard index diagrams of (3.47) are I l 2 and I 13. The modified version 3 2 of (3.12) which arises is - w(i’3)>,

u(:”)

=

4 1 3 )

= w ( i I 3 ) - w(:’3),

w(:”)

from which one obtains the modified version of (3.19), namely, w(;’2)

= u(:”)

w(;~= ~ )u(;I3)

+ + w(iZ3). w(:’3),

273

3. The Integral Representations of the General Linear Groups

Expressions for some of the components with nonstandard index diagrams are U ( y ) = W(i”’>

- w ( y 1 ) = w(:”)

4;31) =,431)

- W ( ; 3 1 ) = ,+,(i23) - 4

U(?”) = w ( ? 1 2 ) - W ( y ) = $423)

- W ( i 2 3 ) = u(:’”, i 1 3 )

-W

( y )

=

-u(;*~),

= -u(:”).

The situation for components with index diagrams of type (e) is illustrated by means of the set of index diagrams

123 4 213 4 312 4 412 3

(3.48)

124 3 214 3 314 2 413 2

132 4 231 4 321 4 421 3

134 2 234 1 324 1 423 1

142 3 241 3 341 2 431 2

143 2 243 1 342 1 432 1

listed in ascending order. There are three standard index diagrams, namely, 123 124 , and i34, in the set. Equation (3.12) assumes the form 4 ’3

- W ( t 2 3 ) = W ( i 2 3 ) - W(?34), u(;’4) = ,4:24> - 3 2 4 ) = W ( y 4 ) - w ( f 3 4 ) , 4;”)= W ( i 3 4 ) - 4 3 3 4 ) = M’(i34) - W ( 3 3 4 ) . U(i23) = W(i23)

(3.49)

While for (3.19), one has W(i23)

(3.50)

W(i24)

w(:”)

+ w(;34), = u<:24> + w ( f 3 4 ) , = 4;”) + 1 4 : ~ ~ ) .

=4

2 3 )

Using Eqs. (3.49) and (3.50) in (3.6), one obtains expressions for some of the components corresponding t o nonstandard index diagrams of the form U(:14)

= wl(:14) - W(;14) = U(:23)

u(2”) = W =

( y )

= W(:24) - M 9 ( ; 3 4 )

+ w ( : ~ -~ )4;”) - 4:”)= u(:23) - w(3 412) - W ( i 2 3 )

4:”)+ W ( f 3 4 ) - u(:’4)

- W(i24)

- U(;34),

- w ( f 3 4 ) = U ( i 2 3 ) - u(i’4).

The reader will observe that even in very simple cases the dimension of the problem gets rapidly out of hand. Nevertheless, the basic procedures are

2 74

4. Representation Theory of Special Groups

quite straightforward. We turn now to the consideration of the matrix representations of GL(V)afforded by one of the standard canonical GL(V) modules. We recall the details of the linear transformation induced on the n-fold tensor product M = V 0 . . . @ V by an element g of GK(V). Let {vl, . . . , vm} be a basis of V and let g E GL(V) be defined by

vig

=

cvja;.

Then the action of g on the extended basis of M is given by (Vil @

. . . 0vi")g

=

1Vj1Uj,il 0. .

=

c

(Vjl

'

0

c vj,

Uj,i"

0. . . 0Vj,)UjlL' . . .

Uj,l".

Let w = (vii0 . . . @ vi,)w(il,. . . , in) be any element of M. Let u = (vjl 0 . . vj,)u(jl, . . . ,j,,) be the image of w under g. Then one has

1

wg = = =

C (vil@ . . . 0vi,)gw(il,

..., in) C ( 1 (vjl 8 . . . 0vj,)ujlil . . . ujnin)w(il,. . . , in) [ 1 \t.(i,, . . . , in)ajlil. . .ujni-](vjl 8 . . .0 vj,,),

which implies that (3.51)

u( j , , . . . ,j,,) =

1 w(i,, . . . , in)ujlil. . . ajnin.

Equation (3.51) can also be written

u(J) =

(3.52)

1 w(1)ug;

in a more compact notation. Both of the equations (3.51) and (3.52) describe the action of g on M in terms of the components of the vector w and those of its transform u = wg. Our problem is to describe the action of g restricted t o a canonical right GL(V)-submodule N ( F , V) of M. This is effected by considering on the lefthand side of Eq. (3.52) only those components u ( J ) for which ( J ) is a standard index diagram of F and by replacing those components il(Z), ( I ) not a standard index diagram, on the right-hand side by their values in terms of components with standard index diagrams. We work out some of the details, starting with Example (3.43) to illustrate the basic idea. (3.53) EXAMPLE. The space M of Example (3.43) is the three-fold tensor product of a space V with basis {vl, v2}. Let g E GL(V) be defined by

v l g = v,u:

+ v2 a:,

v2g

= vlu:

+ v:

u: .

Then one has the action of g on v i 0vj 0vk given by (Vi

0vj 0vJg

=

1(v, 0 v, 0vJaf a i 4,

275

3. The Integral Representations of the General Linear Groups

which leads t o the component equations : u(rst) =

(3.54)

1 w(ijk)afa: a:.

The specific form of (3.54) depends upon the frame F and the corresponding canonical GL(V)-submodule N(F, V). For the case of F1 = . .. , there are four standard index diagrams: 1 1 1 , 112, 122, and 222. The canonical module N ( F l , V) coincides with the space W of symmetric tensors so that the component relations are especially simple. They are

4121) = 4211) = u(l12),

4212) = 4221) = u(122),

with u(l1 I), u(l12), u(122), and 4222) arbitrary. We tabulate these results in a form suitable for later comparisons. See Table (3.55). The columns of Table (3.55) give the expansions of the com(3.55) 111 I12 122 222

111 1

112 121 211

1

1

1

122 212 221

1

1

1

222

1

ponent with index diagram labeling the columns in terms of the components with standard index diagrams labeling the rows. The table is read

u(ll1) = u ( l l l ) , 4121) = u(112), 4221) = u(122),

w(ll1) = w(lll), lC(121) = w(l12), 14221) = w(122),

etc. Making these substitutions into Eq. (3.54), one obtains

u(l11) = "(1 1 I)a:a;a: + w(l12)a:a:a: w(l2l)a:a:a: + w(211)a:a~a~ w(I22)a:a:a: w(212)aIaia: + w(221)a:a:a: + w(222)a:a:a: = w(l1 l)(a:a:a:) w(ll2)(a:a:a: a:a:a: a:a:a:) + w(l22)(a:a:a: + u ~ u : u :+ a:a:a:) w(222)(a:a:a:).

+

+ +

+

+

+

+

276

4. Representation Theory of Special Groups

We now assert that if u(rst) is any component of u corresponding to a standard index diagram rst, then u(rst) = w(I I l)a,!a,la:

+ U;afa:) + w(122)(ara,a, + urasat+ ara,at) + w(222)(ara, a, ). + w(l lz)(afa,la; + a:.: 1 2 2

2 1 2

2 2 1

2 2 2

The reason for this “duplication” is that the index diagram rst on the lefthand side of (3.54) appears as a constant on the right-hand side of (3.54), while the substitutions made for the w(ijk) are obtained from (3.55) and are entirely independent of the index diagram rst. Naturally, the great simplicity of the result depends partly on the fact that we have worked out the case of the symmetric tensors. We turn now to the canonical module N(Fz, V). For the case of Fz = : ’ , there are only two standard index diagrams, namely, l 1 and 12. The analog of Table (3.55) is shown in Table (3.56), where 2 2

(3.56)

11 1

2

11

11 1

12 21 1 1

12 21 22 2 2 1

22 2

-1

12 2

1

- 1

one notes that w(:’) = w(:’) = W($’)= w(;’) = 0 by antisymmetry of the components of w in the columns. One finds that u ( i ’ ) = w(;’)a:a:a$ + w(~1)afa:a: w(:2)alula2 1 2 2 w(122)a,a,a, 2 2 1

+

1 1 2

2 1 1

= w(:’)(a1alaz - a,a,az)

+ w(:2)(a:a:a:

+

2 2 1

- a,a,a,).

In general, one has for u(Y) where rs is a standard index diagram, t

1 1 2

2 1 1

u(:S>= w(:’)(arasat - arasat)

1 2 2 + w(h2)(a,asat

2 2 1

arasat)

for the same reason as before. With these examples in mind, let us indicate a procedure suitable for finding the matrices of the representation afforded by a canonical GL(V)module N(F, V) in the general case. Let M be the n-fold tensor product of an m-dimensional vector space V with itself. Let F be a frame of not more than m rows belonging to n. The canonical irreducible CL(V)-submodule N(F, V)

3. The Integral Representations of the General Linear Groups

277

of M belonging t o F has dimension k equal to the number of standard index diagrams which can be built on F. This number can be calculated by means of Theorem (3.28). The matrix representing an element T of GL(V) can be determined in the following manner. (a) Write down, in ascending order, the complete list D,, . . . , D, of standard index diagrams which can be built on F. (b) Then write down the complete list L,, in ascending order, of all index diagrams, standard or otherwise including D,, which have the same set of indices as D,. Omit from the list all index diagrams which correspond to zero components by antisymmetry. (c) Select the first standard index diagram D,' of F not appearing in the list L,. Write down a second list L,, in ascending order, consisting of all index diagrams including D,' which have the same set of indices as D,'. Again omit any index diagrams which correspond to zero components by anti symmetry. (d) Proceed by induction. If the list Li is completed, select the first standard index diagram Di+,' from those still remaining, if such exists. Write down the list L,+, in ascending order of all index diagrams including Di+,' which have the same set of indices as Di+l'. Omit all index diagrams which correspond to zero components by antisymmetry. (e) Eventually one exhausts the list of standard index diagrams on F. At this time, one has a complete ordered list L, u ... u L , of all nonzero components of tensors which belong to N(F, V). (f) Each list Li , 1 i i i t , contains one or more standard index diagrams, D , < D , < ... < D, , where for simplicity we have omitted a second index on the symbols indicating that they belong to Li . These standard index diagrams are used to label the rows of a table of the form of Table (3.57). The (3.57)

columns of Table (3.57) are labeled with the complete set, El < E, < ... < E, , of all index diagrams belonging t o the list Li . Observe that the standard

index diagrams appear in the columns as well as the rows, but that nonstandard index diagrams appear only in the columns. The entries in the column headed by E j , 1
278

4. Representation Theory of Special Croups

The standard index diagrams are D, = (1 1 l), D, = ( I 12), D, = (122), and D, = (222). The first list I., consists of D, alone. The second list L, consists of (112), (121), and (211). The other two lists L , and L, we leave t o the reader. Note that Table (3.57) assumes the form of Table (3.57") for list L , . (3.57*) 112

112 1

121 211 1 1

(g) The matrix of any linear transformation T on N(F, V) induced by g 0 . . . @ g,g E GL(V), has the general form {T(D j , Di}), where D j and D i range independently through the standard index diagrams of F. The coordinate transformation induced by T has the form involves only index diagrams Each matrix element T ( D j . 0;) to the list of D j . To be specific, one has

Ek belonging

(3.58) where EL occurs in the same list as D j , and the a j n , 1 5 /z 5 s, are found in the j t h row of Table (3.57). The general form of (3.58) is independent of the index diagram D , and depends entirely on the ,jth row of (3.57). The symbol A:?; is formed from the elements of the matrix {t,"}, 1 5 u, L' m, of T with respect to the basis {vl, . . . , v,j of V used t o determine the extended basis of M. It is defined by where { P ~ .. . . , e,,)and ( j , , . . . ,j,J are the index sets of (Ek)and (Di), respectively, obtained by reading from left t o right and top t o bottom in the diagrams. 4. GENERAL REMARKS ABOUT THE REPRESENTATION THEORY OF

CERTAIN MATRIX GROUPS

In this section we discuss various details about what we call the stundurd

matrix groups, a nonstandard terminology introduced in the interest of

later brevity. The standard matrix groups are:

(i) GL,(K) the coniplex generul linear group of all nonsingular, m x m matrices with entries from the field of complex numbers.

4. The Represenfation Theory of Certain Matrix Groups

279

(ii) SL,(K) the complex special linear group which is the subgroup of GL,(K) consisting of all elements with determinant 1. Also called the complex unimodular group. (iii) GL,(R) the real general linear group consisting of all nonsingular, m x m matrices with entries from the field of real numbers. (iv) SL,(R) the real special linear group which is the subgroup of GL,(R) consisting of all elements with determinant 1. Also called the real unimodular group. (v) U(m) the unitary group consisting of all complex unitary m x m matrices. (vi) SU(m) the subgroup of U(m) consisting of all elements of determinant 1. Sometimes called the unitary unimodufar group. (vii) SO(m) the group of all real orthogonal m x m matrices with determinant 1. This is the group of rotations of a n m-dimensional, real Euclidean space X. The subgroups of the complex general linear group introduced above will be referred to as the standard subgroups. Our initial discussion is concerned with the subduced representations of the standard subgroups obtained from the irreducible integral representations of GL,(K). Then we introduce the families of rational, semirational, and semirational integral representations of the standard groups. These families of representations not only play an important role in the general theory, but also exhaust the list of continuous irreducible representation for certain of the standard subgroups. We follow these specific matters with a discussion of a wide variety of facts about matrix groups, Lie groups, and Lie algebras of considerable interest in the applications of representation theory outside of pure mathematics. These facts are discussed more thoroughly and accurately in a number of books from the standpoint of an experienced pure mathematician. Some standard references are : Chevalley (1946), Hausner and Schwartz (1968), Helgason (1962), Hochschild (1965), Jacobson (1962), and Wolf (1967). These books make hard reading for anyone outside of pure mathematics and none-too-easy for many people inside, so that we offer a rigorless presentation of some of the ideas. Since it is almost true that the most difficult thing about a Lie group is its definition, we do not give one. The required discussion of basic topology and geometry tend t o overwhelm many readers long before anything of even remote interest appears on the horizon. We offer a bare skeleton of topological definitions and facts and an equally sparse set from differential geometry. The reader is urged to adopt the attitude that Lie groups, especially matrix Lie groups, are groups which geometrically are very much like surfaces and curves in three-dimensional, real Euclidean space. Naturally, such a treatment will not please everyone. Furthermore, the lack of a precise foundation

280

4. Representation Theory of Special Groups

prevents the inclusion of proofs. Nevertheless, we are hopeful that some will find our discussion a useful entry to a rather difficult area. If they develop a desire for a rigorous presentation there is no shortage of sources. We use the Lie algebra of the three-dimensional rotation group SO(3) as an initial example of a Lie algebra g. The concepts of abelian, nilpotent, solvable, simple, and semisimple Lie algebras are explained. We discuss the Cartan subalgebras of a semisimple Lie algebra and give a very rough sketch of the structure theory. The notions of weights and highest weight of a representation or module of a Lie algebra g are introduced. Examples from the representation theory of S 0 ( 3 ) , S U ( 3 ) , and SU(m) are given to fix the ideas. A connection is established between certain representations of SU(rn) determined by Young’s tableau and the so-called fundamental representations of SU(m). The irreducible integral representations of GL(V) discussed in Section 3 provide integral representations of the group GL,(K) isomorphic to GL(V). Henceforth. we refer to these as integral representations of GL,(K). The irreducible integral representations of CL,,(K) subduce integral representations of the standard subgroups. It is generally true that an irreducible representation T of a group G does not remain irreducible when restricted to a proper subgroup H of G. Consider, for instance, the case of the subgroup H = {l}for any irreducible representation of degree greater than 1. In view of this, the following theorem is noteworthy. (4.1) THEOREM. The irreducible integral matrix representations of GL,(K) remain irreducible when restricted t o the complex special linear group SL,(K), the real general linear group GL,(R). the real special linear group SL,,(R), the unitary group U(m),and the special unitary or unitary unimodular group SU(m). The matrix representations afforded by different canonical modules N(F, V) sometimes give the same representation when restricted t o SL,(K), SL,(R), and SU(m). One obtains all distinct integral representations of these last three by considering frames with at most m - 1 rows.

(4.2) REMARK. One should note that the m-dimensional rotation group SO(m), sometimes called the real special unitary group, is not contained in the above list. This absence suggests the introduction of different methods for discussing the representation theory of SO(m). There is a broader class of irreducible representations of the general linear group which we mention quite briefly. The natural generalization of the integral representations of the group GL,(K) are the rational representations T i n which the elements of the matrix T(g) corresponding t o g E GL,(K) are rational functions in the elements of g. A still larger class of representa-

281

4. The Representation Theory of Certain Matrix Groups

tions are those in which the elements of T(g) are either rational or integral functions of the real and complex parts of the elements of g. To make this last idea quite definite, let o l l is,, oI2 it,, g = 021 it,, oZ2 it,,

It

+ +

+ +

l

denote an element of GL,( K ) . The semirational representations of GL,(K) are those in which the elements of T(g), g E GL,(K), assume the form A o i j tij)/q(oij> tij), where p and q denote polynomials in the eight variables {ol . . . , t,,}. The semirational integral representations are those in which the elements of T ( g ) are of the form p ( a i j ,s i j ) , where p is a polynomial in eight variables. The definition in the cases of GL,(K) and the standard subgroups should be clear to the reader. These kinds of representations of GL,(K) generally turn out t o be completely reducible and to have irreducible components which can be expressed in terms of the irreducible integral representations of Section 3 and the determinant function. We have the following two theorems about finite-dimensional representations. 9

(4.3) THEOREM. Every semirational representation of SL,n(K) is a semirational integral representation. All such representations are completely reducible. (4.4) THEOREM. Every continuous representation of the group SL,(R) is integral and completely reducible. We are particularly interested in the group SU(m) which plays an important role in the applications of representation theory to physics. The continuous representations of SU(m) are given by the following theorem.

(4.5) THEOREM. All continuous representations of the group SU(m) are integral. The continuous irreducible representations are obtained from the canonical GL(V)-submodules N(F, V) for frames with not more than m - 1 rows. One development of the representation theory of SU(m) and SO(m) sometimes proceeds along lines quite different from those discussed above. In particular, their representation theory as well as that of the other standard groups can be investigated with the methods of topology and differential geometry. A detailed study of such methods is beyond the scope of the present book. Nevertheless, we will make a brief survey of the application of topological and geometric ideas in representation theory. Unfortunately, our discussion must begin with a rather long list of unmotivated, but necessary definitions. [The reader who finds the list too tiresome can skip and return later if so inclined.]

282

4. Representation Theory of Special Groups

Let S be a nonempty set. A topology z on S is a dis(4.6) DEFINITION. tinguished family of subsets of S such that

uuEx

(i) the set S and the empty set @ are members of z; (ii) if each of {U,,}, G E E , belongs to z, then U, belongs t o T ; (iii) if U and V belong to 5 , then U n V belongs t o z. The subsets or elements of the topology z are called open sets of S. A family 8 of subsets of S is said to be a hasis of the topology z if every nonempty set L' of z is the union of sets from 8. The familiar rn-dimensional real Euclidean spaces RE(/??)are given a topology z by taking as a basis 8 of z the collection of all c-spheres S,(p). F: > 0, p E RE(rn), where S,(p)

= {x E

RE(rn): (1 x - pII < E } .

Here I(x - pII denotes the Euclidean distance from x t o p in any real Euclidean space RE(iii). A szthhusis of a topology z on a set S is a collection 3 of subsets of S such that every U E z is the union of sets, each of which is the intersection of a finite number of sets of I?. We give only examples of bases of a topoIogy since bases are more important for us than subbases. The set 8 of open circles C = {x E : IIx - xo 11 < E ] , xo E E > 0, together with the empty set @ is a basis for the standard topology of the two-dimensional real Euclidean plane 5Q. A subset 0 of the plane '1' is open in this topology if and only if each point x of 0 is contained in some open circle C which itself is contained in 0. The set 8 of open spheres S = {x E X : I/x - xo I/ < E } together with @ is a basis for the usual topology of three-dimensional real Euclidean space X. A set 0 of X is open if and only if either it is @ or the union of open spheres. The nonempty open sets of the standard topology of any /n-dimensional real Euclidean space RE(n7) are those containing an open sphere about each of their points. Most of our topological considerations deal with real Euclidean spaces. A topology is usually introduced on a nonempty set S in order to define notions of nearness and continuity. The basic concept is that of a continuous m a p f ' : S W of one topological space S into another 12.:

v,

--f

(4.7) DEFINITION. The map f : S + W is continuous if and only if f - ' ( O ) is open in S for every open set 0 of W. A bijection , f : S ---t W is a h'omeonzorpiiistii iff f and f are continuous.

-'

(4.8) REMARK.Definition (4.7) is completely equivalent t o the standard one of elementary calculus. This states that the map f :RE(n) -+ RE(rn) is continirous ut the point x E R E ( n ) if and only if given E > 0, there exists 6 > 0 such that 11 f ( y ) -f(x)II < c whenever / / y - x/I < (5. The functionf: RE(n) + RE(r77) is said to be continuous if and only if it is continuous at every point

4. Zhe Represenration Theory of Certain Matrix Groups

283

x E RE(n). Now let f(x), x E RE(n), belong t o any open set 0 of RE(m). Then some open sphere

is also contained in the open set 0. By continuity off at x, there exists 6 > 0 such that if jly - x/I < 6, then Ilf(y) -f(x)II < E. This shows that the sphere S,(x) = { y e RE(n) : I1y - xI1 < S } belongs tof-'(O). Consequently, f -'(O)is open since it contains a spherical neighborhood of each of its points. Thus the calculus definition implies (4.8). The converse is left to the reader. A nonempty subset V of a topological space S inherits a topology from the including space. This topology is equivalent to the one with the " smallest number of open sets" with respect to which the inclusion map i : V - + S is continuous where i(x) = x, x E V . (4.9) DEFINITION. The open sets of the relutioe topology of Y (as a subspace of the topological space S ) consist of all subsets of V , each of which is the intersection with V of an open set of S. (4.10) REMARK. The subset V with this topology is called a subspuce of S. There are various reasons why this rather strange-looking definition is adopted. Consider some familiar examples of topological spaces. Let '$ be any two-dimensional plane contained in the three-dimensional real space X. The basic open sets of X are open spheres S, each of which has either an open circle or the empty set for its intersection with 'Q. The induced or relative topology of '$ as a subspace of X agrees with its usual standard topology. Let V be the boundary

v = {x E x : llxll = 1)

of the closed unit sphere in X. The intersection of the open spheres in X with V are again the right objects to define the usual topology of V. Let V be the restriction of a continuous function f:S -+ W to some subspace V of S. If 0 any open set in W, one has ( f l V)-'( O) = f - ' ( O ) n V , an open set in V. Thus the restriction f l V of the continuous function f on S is a continuous function on V with its relative topology.

fl

Several other notions from topology play an important role in our subject. We state additional definitions, some of which are somewhat special (not the general definition), but suitable for our purposes. (4.1 1) DEFINITION. A neighborhood of a point p in the topological space S is any open set of S containing p.

(4.12) DEFINITION. An urc (closed arc) in a topological space S is the continuous image of an open (closed) interval of real numbers.

284

4. Representation Theory of Special Groups

(4.13) DEFINITION. A siniple closed curz’e in a topological space S is any homeomorphic image of the boundary of the unit disk in the two-dimensional Euclidean plane.

A topological space S is connected if any two distinct (4.14) DEFINITION. points a and b of S can be joined by a closed arc in S. (4.15) DEFINITION. A topological space S is locally connected if given any neighborhood N of a point p of S there exists a connected neighborhood C of p which is contained in N . (4.16) DEFINITION. Let S be a connected, locally connected topological space. Suppose that any simple closed curve C in S can be continuously deformed t o a point within S. Then S is called a simply connected space.

(4.17) EXAMPLE. The unit cube in RE(m) is a simply connected space for rn > 0. All the real Euclidean spaces are simply connected. The boundary of the unit circle in the plane is a connected, locally connected space which is not simply connected. We need several other topological notions which will not be precisely defined. A topological space S is compact means among other things that every continuous functionf: S -+ R from S into the real numbers R assumes its largest and smallest value at some points of S. A topological space S is locally compact means that every neighborhood N of a point p E S contains a compact set C which in turn contains a neighborhood W ofp. Letf: S + W be a continuous map from the topological space S onto the topological space W . There exists an equivalence relation R on S defined by xRy if and only if f(x) =f(y). This relation partitions S into equivalence classes where [XI denotes the class containing x. There exists a bijectionf’ from the set S’ of equivalence classes onto W definedf”~] =f(x). The set S’ can be given a quotient topology such that ,f’ is a homeomorphism. Let f:J + RE(m) be an arc in RE(nz) with domain the open interval J of real numbers. (4.18) DEFINITION. The arcf is dixerentiahle at thr point x E J if and only if limy+x[f(y) -f(x)]/[y - x] exists. The arc f is diflerentiable on J if it is differentiable at each point of J. Let {vl, . . . , v,} be a basis for RE@). Then for x E J,

+ . . . +f,(x)v,n where each element of the set (f,}, 1 i i 5 nz, is a map from f(x)

=f1(X)V1

?

J to the real numbers. It is a simple matter t o prove that the arc f is differentiable if and only if each of the components fi, 1 5 i 5 rn, is differentiable.

285

4. The Representation Theory of Certain Matrix Groups

These very general ideas become applicable t o the standard matrix groups by an embedding of them as subsets of real Euclidean space RE(n7) for a suitable value of m. The complex general linear group GL,(K) is a subset of the algebra K, of all complex matrices of the form { a j j } . 1 I i, j 5 m, where each a i j has the expression a j j = a i j + i.rij with a i j and s j j real numbers. There exists a bijection f :K,,, -+ RE(2m2)defined by

f [(aiJl

= (01

1 7

71 1,

.. ., ~ m r n rrnm), 7

In the case of K 2 , the map assumes the form

We identify K , with its image in RE(2m2),noting in particular, that f embeds GL,(K) as a subset of RE(2m2).Indeed, GL,(K) is all of RE(2m’) except for a hypersurface consisting of those elements of K , of zero determinant. We make a few remarks about hypersurfaces below. I n particular, GL,(K) is an open subset of RE(2m2) with every g E GL,(K) contained in an open 2m2-dimensional sphere consisting entirely of points of GL,,( K ) . Roughly speaking, GL,(K) is similar to the subset of RE(2m’) which remains after removing a plane, that is, a hyperplane. However, there is an important difference in that GL,(K) is a connected subset of RE(2m2).Locally GL,(K) is like RE(2m2)and most of the usual concepts of real Euclidean space are fully meaningful. We assume of the reader a good intuitive grasp of the concepts of arcs and tangent vectors, surfaces and tangent planes, and the like in three-dimensional space and a willingness to accept the extensions of these ideas t o higher dimensional spaces without benefit of full discussion and proofs. Fortunately, our present situation is different from that incurred in many places in modern analysis where frequently one is mainly concerned M ith how badly one’s intuition goes astray. The area in which we work is one in which the development has been along lines agreeing with intuitive notions. We consider geometric objects which are like surfaces and which we call h-surfaces, meaning higher-dimensional surfaces. Generally, such h-surfaces arise as the solution sets of one or more algebraic equations in 2m2 unknowns. To clarify the idea, consider the ordinary sphere in three-dimensions which is the solution set of the algebraic equation (4.19)

x2+y2+z2=1.

Other classical surfaces such as ellipsoids and hyperboloids are the solution sets of similar quadratic equations. Given a second equation such as (4.20)

(x

- 1)2

+ yz + z2 = 1,

286

4. Representation Theory of Special Groups

the reader notes that the simultaneous solution set of (4.19) and (4.20) is a circle. again a familiar geometric object We are interested, generally speaking. i n the solution set of a family of h- equations in 2rn2 unknowns of the form (4.21)

/ I ~ ( . Y ~. .. .

.

.xZrn2)

= 0.

1
where the function I < , j 5 k . is algebraic. The solution set is generally ( 2 m 2 - /<)-dimensional when the set of functions {pi: is independent, a concept we d o not try t o make precise. The concepts of arc and of a tangent vector t o an arc pass from three t o 2/n2 dimensions without loss of intuition or meaning. A simple closed curve in RE(2/n2) is the homeomorphic image of a circle in the two-dimensional plane. The arcs which arise in our discussions have tangent vectors which turn continuously as one progresses along the arc. The real general linear group GL,,,(R ) is an h-surface which is the subset of the complex linear group GL,,(K) obtained as the solution set of ni2 real equations. namely. the set of equations asserting that the imaginary parts of all elements of a matrix g E GL,,( K ) are zero. Since GL,,,(K ) has real dimension 2m2. it follows that the h-surface GL,,(R) has dimension m 2 , a well-known fact. The unitary subgroup U(n7) of GL,,,(K)i s determined by a set of equations which assert that { u i , ! . 1 < i.,j 5 177. is an element of U(n7) if and only if (4.22)

uiiiiki = 0,

I Ij < I< I m.

N ~ ~= L I~ . , ~

I I j I 111.

and (4.23)

There are 1 4 1 7 7 - 1)/2 complex equations in the set (4.22) and. effectively. 177 real equations in the set (4.23). Both sets together determine i n 2 independent real equations that must be satisfied in order that a matrix { a j j )of GL,,(K) belong t o the subgroup L"(n7).Thus C'(n7) is an h-surface of dimension m 2 . One can also argue that the special unitary group SL'(n1) is an h-surface of dimension / n 2 - I . These facts about 11-surfaces have a familiar and useful analog. Recall that the solution set S of a family of k independent real linear equations in n real unknowns. (4.24)

2 u j j x j= 0,

1ii

k < n,

can be described in several ways. One such way is to state there exist n - k parameters ( i , .. . . . I., -kj such that each point {xi},I 5 i i n, of S is given by (4.25)

xi= ~bjj?.j, I I j l n - k ,

I
4. The Representation Theory of Certain Matrix Groups

28 7

The set S has dimension n - k in such cases. The parameters {;vi} assume all possible real values and determine each point on S exactly once. Similar statements can be made about the nature of solution sets which are h-surfaces of GL,(K). The principal differences are that the equations replacing (4.25) must be more complicated and that one can be sure of nice solutions only in a suitable neighborhood of a point on the h-surface. We illustrate the basic idea by considering an element g E U(m). There exists an open neighborhood N ( g ) of g in GL,,(K). 6 > 0. and a set ( f i j } , 1I i , j 5 m2, of functions such that each element ( a i j ) of C ! ( m ) contained in N ( g ) is determined exactly once by

(4.26) for those

(4.27)

a I-J .= f i j ( ) . , .

i=

.

. . . I .,,,2 ) .

. . . . I.,,.) with

l i , - Xio\

< 6,

I I i , j 5 tn2,

I I i i m2.

Here { f i j ( x o ) ] 1, 5 i. j 5 1 1 7 ~ .is the matrix of g. Furthermore. each of the functions f i j , 1 2 i, j 2 i n 2 , is an analytic function of the real variables { I b l . .... in the sense that it has a convergent power series expansion about Lo. In particular. each f i j is infinitely differentiable with respect to the set ( I v l , . . . , I.,,,,) of parameters. By fixing all except one of these. say of these with their value at one obtains a curve G j in the space GL,(K) which passes through the point g and which has the tangent vector H j at g = {Aj(%')}. The tangent vector H j can be regarded as an element of K,,, , but not always as one of GL,,(K), in the same way that the tangent vector to an arc G in the three-dimensional real Euclidean space X is usually taken to be an element of X.

x".

(4.28) EXAMPLE. The real ortitogonu1 group O(m) is the hubgroup of real matrices belonging to the unitary group L'(n2). The special real orthogonal group SO(m) consists of those elements with determinant 1 in O(n7). Geometrically, SO(m) is the group of linear transformations on the real Euclidean space R E ( m ) which leave invariant both the length and orientation of vectors. This group is commonly called the in-dimerisional rotntion group. The group SO(3) is the famiiiar group of rotations of three-dimensional space. Not only is it a good example of the ideas under consideration. but also its representation theory is important in the applications to quantum mechcnics. An element { a i j ) ,1 5 i , j 2 3, of GL,(K) belongs to SO(3) if and or:ly if it has real elements which satisfy the equations: (4.29) and

(4.30)

x:ajiaji= i ,

1 < , i s 3.

288

4. Representation Theory of Special Groups

Thus O(3) is a subgroup of the nine-dimensional real general linear group GL,(R), and its components satisfy six real equations. We conclude that O(3) is three-dimensional. Each element g E O(3) has det(g) either I or - 1. Thus SO(3) is the part of O(3) with det(g) equal to 1. Since this is not an independent relation, the dimension of SO(3) is also 3. This dimension can be obtained by geometric considerations. Every rotation R in the threedimensional space X is determined by specifying an oriented axis and a magnitude 4 of rotation, Consequently, the three-dimensional rotation group SO(3) can be placed in correspondence with the sphere 6of radius n: drawn in Fig. (4.31). Let f : G + SO(3) make correspond to each vector belonging (4.3i)

t

X

to 6 the rotation R whose directed axis is along the direction of and whose magnitude 4 i s the length of Any rotation R with magnitude 4 greater than i( coincides with a rotation R' with magnitude 4' not exceeding n about the same line, possibly differently orientated. This shows thatfmaps G onto SO(3). The niapfis also one-to-one except for 2 of length n. For these, f ( f ) and are the same rotation. The sphere 6 bears its relative topology as a subset of X and the group SO(3) bears its relative topology as a subset of GL,(K). Since a rotation changes little when its axis is barely tilted and its magnitude slightly altered. the map f is seen intuitively to be continuous. When one identifies antipodal points, such as a and a' of 6in Fig. (4.31), to obtain a new set E',the induced map f ' : S' + SO(3) arising from f is a homeomorphism of G' (in the quotient topology) onto SU(3). Thus SO(3) i~ topclogically like the sphere 6 with antipodal points identified. This identification doe\ not change the local nature of the sphere 6. Consequently, a small neighborhood in SO(3) is topologically like a small neighborhood of KE(3). However, the global nature of RE(3) is quite different from the global nature c f SO(3). To point out two differences: RE(3) is locally compact, but not compact while SO(3) is compact. The space RE(3) is simply connected.

x.

f(-x)

289

4. The Representation Theory of Certain Matrix Groups

Recall this means that RE(3) is connected, locally connected, and any simple closed curve C in RE(3) can be continuously deformed to a point. On the other hand, SU(3) is not simply connected. A simple closed curve in SU(3) which can not be continuously deformed into a point is the image underf’ of the segment aa‘ of Fig. (4.31). This segment is a simple closed curve in S’. If one attempts t o move the point a = a’ from its position on the boundary” the curve breaks. Otherwise, one can not deform it into a point. However, both RE(3) and SO(3) are connected. According to Eq. (3.9), Chapter 3, every rotation R of three-dimensional real space X has a matrix of the form “

0 -sin a cos a

0 cos a 0 sin a

with respect t o a suitably chosen basis. To obtain this form, the x-axis is taken to be the axis of rotation of R and c( the radian measure of the magnitude of the rotation. We occasionally allow ourselves the convenience of confusing the idea of a linear transformation and its matrix in the remainder of this section. This often makes the discussion less awkward, and lets the reader supply the required interpretation. In particular, we confuse GL,(K) and GL(V). The rotation R has a trace given by 1 + 2 cos a which suggests the following lemma. (4.32) LEMMA. Two rotations R , and R2 of the three-dimensional real space X are conjugate if and only if they have the same trace.

The proof is omitted. Note that the set of all rotations about any fixed line of the real threedimensional space X is a subgroup of SU(3). By proper choice of axes and notation, the group H , of rotations about the x-axis consists of all rotations of the form 1

0

0

sin 2,

cos lL1

This is a basic example of what is called a one-parameter subgroup of a Lie group. As such, it can be considered as a map H I : R SU(3) which is a homomorphism of the additive group R of real numbers into the group SU(3). The general definition is the obvious analogue. We see below that the one-parameter subgroups play an important role in the general theory. There --f

290

4. Representation Theory of Special Groups

are, of course. one-parameter subgroups H 2 and H , of rotations about the y-axis and the z-axis. These are described by

HZO”,) =

1

cos)i., -sin ?,.

siY2 0 cos 2,

and 0

I/

1

The one-parameter subgroups play a significant role since the elements in a neighborhood of the identity of many important groups can be written as the products of elements from a finite family of one-parameter subgroups. To see such a factorization in the case of SO(3), let R denote any element of SO(3). Let {v,, v 2 , v3} be an orthonormal basis of the real three-dimensional space X, and let R be defined by Rvi = u i , 1 i i 5 3. It follows that the set { u l , u 2 . ujl is also an orthonormal basis of X . (i) There exists an element H,(A1’) rotating u, into the (v,, v,)-plane with - n < i,’ 5 0. ( i i ) There exists an element H 2 ( A 2 ’ ) such that H2(?.2‘)Hl(Al’) rotates u3 into vj with -277 < A’ 5 0. (iii) There exists an element H3(A3’) such that H,(n3’)H2(~2’)H,(Alf) rotates u, into v, and u2 into v2 with -27t < A,’ 5 0.

One has H 3 ( I . , ’ ) H 2 ( ? ~ , ’ ) H ~ ~ A 1 ’= ) Uvii .

1I i I 3,

so that R - ’ = lf3()~3’)H2(A2‘)H,(A,’)r from which it follows that R can be written in the form R =~10”1)~2@2W3(~3)~

(4.33)

where 0 I i,, < n. 0 5 L2 < 271,O i A, < 27t with 2, = -Ai‘, 1 5 i i 3. The matrix o f fi with respect to a set of coordinate axes through v,, v2, and v3 is given in Fig. (4.34).

(4 34)

!I

co\

A,

XI

cos A,

\in A,

sin A, sin ‘lsiii A, $111 A, - coy A, sin i w

A2

cos A3 cos A,

A, sin A 3 cos A , cos A, sin A , sin A, sin A, sin A, cos 1 cos A, sin A, sin A,

-cos

-sin A, cos A,

cos A, cos A,

Thu\ n e find that every element of SO(3) can be written as the product of element\ from the subgroups H , , H,, and H,. Furthermore, given any neighborhood Ny,) of the element go = H1(A,’)H2().2O)H,(~3*) of SO(3),

291

4. The Representation Theory of Certain Matrix Groups

zio]

there exists a 6 > 0 such that the set of f with I f i < 6 determines, exactly once, each element g(x) of a neighborhood ” ( g o ) contained in N(go). Here

.dx)= H1(;il)H2(n2)H3(A3). Let g(x‘) be another element in ”(go). Then it is clear from the form of (4.34) that g ( f ) g ( z ’ )is a matrix whose elements are analytic functions of the components of and i’. Let J1 be the open interval (5 : lLlo - S < 5 < iIo + S}. Then there exists a differentiable arc f,: J1 + SO(3) defined by (4.35)

f i(4) =

(5>H2(n,0)H3(ju30>.

We refer to the a r c h as a A,-path through g o . One can define ;.,-paths through g o , 1 < i 2 3, in an analogous fashion. The tangent vector to the path fiis given by

4 /dt= Wl ( i ” ) / ~ ( 1 H 2 ( ~ 2 ° > ~ ~ ( ~ 3 0 ) , These concepts prove most useful in a neighborhood of the identity element of SO(3) which has the coordinates = 0. The expressions for the A-paths fi, f 2 , and f3 in a suitable neighborhood of the identity in SO(3) are 1 0 0 0

sin

5

cos 5 0 -sin( cos 5

cos

5

sin

t

0 cost -sin

5

0

At the identity of S0(3), these arcs have the tangent vectors

292

4. Representation Theory of Special Groups

One notes that each of these three tangent vectors is a skew-symmetric matrix. that is, its transpose is its negative. Furthermore, any real linear combination of the three. for example,

I

0

-a3

nz

Jj.

-a, -a, 0 is also skew-symmetric. Conversely, any real skew-symmetric matrix is a real linear combination of A , , A , , and A , . In this sense, the tangent plane eo(3) at the identity of SO(3) can be identified with the set of all skewsymmetric. 3 x 3 real matrices. This set of skew-symmetric rnatrices is a linear space over the real numbers, but it is not closed under ordinary matrix multiplication. It proves to be a most fruitful idea t o introduce a binary operation [ , ] on the real linear space eo(3). If A and B are two real, skewsymmetric matrices in 9 4 3 ). then one defines n,A,

f

02A2

f

a,A, =

0 a,

a3

[ A , B ] = A B - BA,

(4.36)

where A B and BA denote the standard products of the matrices A and B. Note. i n particular. that [ A , B ] is skew-symmetric whenever A and B are skew-symmetric. This product [ A , B ] i s called the Lie product of A and B. The real linear space 543) together with the Lie product [ , ] is called the Lie ulgc&a of the special orthogonal group SO(3). One finds by direct applications of the definition that (4.37)

[ A , B] = -[R. A].

[A. B

{ ' [ A . BJ = [ P A , B ] = [ A , p B ] .

+ C ] = [ A . B] + [ A , C],

[A

+ B. C ] = [ A , C ] t [ B , C],

where A . B. C E so(3) and p is any real number. Thus most of the familiar laws of algebras are valid with one special exception. The usual associative law fails and is replaced by a more complicated rule, (4.38)

[ [ A . B], C]

+ [ [ B ,C]. A ] + [[C. A ) , B] = 0.

which is usually referred t o as the Jacobi idcnritj.. (4.39) DEI-'INITION. A rcwl Lic algrhra A is a vector space over the real numbers for which rhere is defined a Lie product [ , ] such that the rules of (4.37) and (4.38) are satisfied. The theory of Lie algebras is highly developed. We refer the reader t o Kaplansky (1963) for an elegant introduction and t o Jacobson (1962) for details. Lie algebras admit of a very detailed classification in many instances. ification is of great interest in the study of Lie groups of which GL,,(K)and many of its subgroups are particular instances.

4. The Representation Theory of Certain Matrix Groups

293

The discussion of the group SO(3) is special only in the details. Each of the standard groups has associated with it a real Lie algebra. This Lie algebra is obtained by examining a neighborhood of the identity of the group in question and determining a family of one-dimensional subgroups which play the role of H I , H 2 , and H , in the case of SO(3). The results are as follows: (a) The Lie algebra of the complex general linear group GL,,,(K)is the set gI,(K) of all m x m complex matrices. (b) The Lie algebra of the real general linear group GL,(R) is the set gl,(R) of all in x m real matrices. (c) The Lie algebra of the complex special linear group SL,(K) is the set 51m(K)of all m x m complex matrices with trace zero. (d) The Lie algebra of the real special linear group SL,(R) is the set sI,(R) of all m x m real matrices with trace zero. (e) The Lie algebra of the unitary group U(m)is the set ~ ( mof) all skewHermitian complex m x m matrices. (f) The Lie algebra of the special unitary group SU(m) is the set sii(m) of all skew-Hermitian m x m complex matrices with trace zero. (g) The Lie algebra of the special orthogonal group SO(m) is the set so(m) of all skew-symmetric m x m real matrices. The Lie algebras discussed so far arise from h-surfaces in real Euclidean spaces. Consequently, the scalars involved are the real numbers. However, Lie algebras exist over any field. We first discuss the case of real Lie algebras and then turn t o complex Lie algebras where various problems are simpler. Most definitions given for the real case extend directly to any field and are not repeated. The determination of the one-dimensional subgroups of a matrix group may present difficulty for some of the various subgroups of the complex general linear groups. However, if the Lie algebra of a subgroup G of GL,(K) is known from other considerations, it is easy to specify the one-dimensional subgroups of H and G. The method employs several results on matrices and linear differential equations which we discuss briefly. Let A be an n x n complex matrix. Then exp(A) is (4.40) DEFINITION. defined by the infinite series (4.41)

exp(A) = 1 + A

+ A 2 / 2 !+ . . . + A " / n ! + . . . ,

which is the same series, of course, used to define exp(x) for x a real or complex number. The convergence of (4.41) is most easily proved by use of a norm I] I/ on

294

4. Representation Theory of Special Groups

gl,,(K) which determines a topology equivalent to the Euclidean topology on gi,,(K). The obvious candidate for the norm is given by

(4.42)

IlA/12 =

2 luijlz =

-y{cTij2

+

Zij2>,

so that the norm (IAlj of A is its Euclidean distance fron the zero matrix. The &-spheresof this norm S,(A) = ( M

E gI,(K)

: (1 M

-

A 11 < E )

are the spherical neighborhoods of the Euclidean topology so that the topology induced by the norm /I I/ is the same as the Euclidean topology on gi,,,(K). The norm 11 11 prokides a convenient working tool because of the fundamental inequalities IIA

+ Bll I!I4+ IIBII

and

IlABlI

s 114 IIBII.

These rules extend t o the cases of IZ summands and n factors by induction. The approximating sums of (4.41) are of the form

s, = 1 + A + . . . + A " / n ! , Then. for i < j , one has [ISi- S j / (= /ISj - Sill = I \ A i + l / ( i + I)!

- a i + l /(i+ < I)!

0 < n.

+ . - .+ A j b ! l (

+ ... + u j / j ! ,

a = I/AIl.

Since this last s u m tends to 0 as i tends t o infinity, it follows that (4.41) converges t o an ti x IZ matrix. Many of the usual rules for the numerical exponential function remain valid: exp(nA) where 1 is the

= (exp(A))" II

x

ti

and

exp(A) exp( - A )

identity matrix and 0 is the n x

(4.43)

exp(A

IZ

= exp(0) = 1,

zero matrix. The rule

+ B ) = exp(A) exp(B)

holds when A a i d B are commuting matrices; otherwise, certain difficulties arise which are considered in a more complete treatment. Let t

E(X E

then the series exp(tA)

=

1

R 1 --b < x < b},

+ tA -t. . . 3. t"A"/n! + . . .

converges uniformly as a function o f t , and d exp(tA)/dt = A exp(tA).

295

4. The Representation Theory of Certain Matrix Groups

One knows from the theory of linear differential equations with constant coefficients that any matrix equation of the general form dfldt

= Af

has a solution of the form

f = C exp(tA), where C is a constant matrix. One notes that given any A E gI,(K), thenf(t) = exp(tA) is a differentiable arc in GL,(K) whose tangent vector at the origin is the element A . Furthermore, one has by (4.43) that f ( t + t’) = exp(tA

+ t ’ A ) = exp(tA) exp(t’A) =f ( t ) f ( t ’ ) .

Thus the set {exp(tA) : - co < t < a} is a one-dimensional subgroup of GL,(K) with tangent vector at the origin equal to A . It can be shown that all of the one-dimensional subgroups of GL,(K) have this form. More generally, if A is an element of the Lie algebra of G, any of the subgroups of GL,(K) under consideration, then one obtains a one-dimensional subgroup H of G by the process indicated. All one-dimensional subgroups of G arise in this manner. Consider the three tangent vectors at the origin, A , , A , , (4.44) EXAMPLE. and A , , of the special orthogonal group SO(3). Then one finds that H i ( t ) = exp(tA,).

To be specific,

1:

H , ( t ) = 0 cos t sip,

1IiI 3.

0 -sin t c ost

1.

We turn to a brief explanation of our interest in these matters. The complex general linear group GL,(K) and its subgroups under discussion are all examples of Lie groups. We will not give a formal definition, but remark that Lie groups are similar to the standard groups. However, the technical problems in topology and differential geometry become substantially deeper for Lie groups in general. Nevertheless, many basic concepts for general Lie groups are analogous to those for matrix groups. In particular. each Lie group G has associated with it a Lie algebra g which is the tangent space t o G at the origin. The algebraic properties of the Lie algebra g strongly influence those of the associated Lie group G. Any two simply connected Lie groups G and G‘ with isomorphic Lie algebra g and g’, respectively, are themselves isomorphic. The relationship remains strong when the Lie group G is connected, but not simply connected.

296

4. Representation Theory of Special Groups

If G is a connected Lie group with Lie algebra g, then there exists a simply connected Lie group G' whose Lie algebra g' is isomorphic to 9. Furthermore, G is a homomorphic image of G'. Actually, G' contains a discrete central subgroup N such that G is isomorphic to G I N .

(4.45) EXAMPLE. The real numbers R under addition and the unit circle C = { z E K : lzl = I} under multiplication are familiar examples of Lie groups. each of which has the real numbers as its Lie algebra. The group R is simply connected while the group G is only connected. The normal subgroup N such that C is isomorphic to R / N can be taken t o be N = {x E R : x = 2nn, IZ EZ}. The groups SU(2) and SO(3) share an analogous relationship. The discrete central subgroup N of S U ( 2 ) is {l, -1} where 1 denotes the 2 x 2 identity matrix. There exists a homomorphismf: SU(2) + SO(3) with kernel N which is a homeomorphism on a sufficiently small neighborhood W of 1. More generally, let G be a connected Lie group with Lie algebra g and let G' be a simply connected Lie group with Lie algebra g' isomorphic to g. Then there exists a neighborhood W of the identity 1' in G' and homomorphismfof G' onto G such that the kernel N offis a discrete central subgroup of G' meeting W only in 1'. Furthermore, the homomorphismfis a homeomorphism on the neighborhood W . A map such a s j i s called a local isomorphism, and a group such as G' is called a universalcoaeringgroup of G. If H any other connected Lie group whose Lie algebra lj is isomorphic to g, then G' is also a universal covering group of H and there exists a homomorphism y : G' H such that for some neighborhood V of I' g is a local isomorphism. A full discussion of this relationship between Sb'(2) and SO(3) is not difficult. However, we must refer the reader t o Gel'fand and Sapiro (1952, p. 213) for it. These observations support the rather vague statement that the nature of a Lie group G in the neighborhood of its identity is largely determined by its Lie algebra. The close relationship is shown for GL,(K) and its standard subgroups by means of the exponential map exp : gl,,,(K) + GL,(K) defined for A E gI,!,(K) by --f

exp(A) = 1

+ A + . . . + A"/n! + . . . ,

and for Lie groups in general by a similar but technically more complicated function. These exponential maps are homeomorphisms of some neighborhood of the 0 element of the Lie algebra 9 onto some neighborhood of the identity I of G. The structure theory of many classes of Lie algebras is known in detail. This is true especially for the Lie algebras of the standard matrix groups. Moreover. the finite-dimensional representation theory of the Lie algebras of these groups is well understood.

4. The Representation Theory of Certain Matrix Groups

297

(4.46) DEFINITION. A finite-dimensional representation t of a real Lie algebra g with Lie product [ , ] is a mapping with domain g and range Hom(V, V) for some r-dimensional real vector space V such that

(4.47)

t(au

+ pv) = at(u) + Pt(u)

and

(4.48)

f([u, v1) = [t(uh t(v)17

where [t(u), t(v)] is t(u)t(v) - t(v)t(u), the usual additive commutator, a, are real numbers, and u, v are elements of g.

fl

Rather than a map t from g into the associative algebra of linear transformations Hom(V, V) one can define a map into the isomorphic algebra R, of r x r real matrices. In this case, one has a matrix representation of the Lie algebra g. There is also the usual technique of rep!acing a representation by a module and the converse.

(4.49) DEFINITION. Let M be a finite-dimensional real vector space. The space M is said to be a module for the real Lie algebra g if (i) there is a left multiplication xm defined for elements m of M by elements x of g; (ii) x(crm, + pm2) = cr(xm,) + p(xm2) for x E g , m , , m2 E M, and 2, fi real numbers; (iii) [x, y]m = x(ym) - y(xm) for x, y E g, m E M . We should be familiar by now with the fact that the concepts of representation and module are mostly different ways o f looking at the same thing. Let G be a standard matrix group with Lie algebra g . Denote by env(G) the real enveloping algebra of C consisting of all real linear combinations

+ ... + a n g n ,

"191

a iE R , g i E G, i _< i _< n. Then env(G) is a real associative algebra which is a closed subspace of the algebra R, of all real M x m matrices. There exists a neighborhood W of the 0 matrix in g such that exp maps W homeomorphically onto a neighborhood of the identity 1 of G. Given A E g, there exists 6 > 0, such that exp(xA) E W when 1x1 < 6. This means that

(exp(xA)

-

l)/x

E

env(C)

when 1x1 < 6. Since env(G) is closed, one has

A is an element of env(G).

=

lim (exp(x.4) - l ) / x

x-0

298

4. Representation Theory of Special Groups

Any finite-dimensional representation T : G -+ GL(V’)extends in a natural manner to a representation of the associative algebra env(G). We denote the extended representation also by T. Define t : g Hom(V’, V’) by --f

t(A)

for any A

E

=T[

lim (exp(xA) - I)/x] = T(A)

x-+o

g. Since T is a representation of env(G), one has

t ( [ A , B ] ) = [ ( A B - B A ) = T(AB - BA) = T(A)T(B)-

=

[W), W ) I = [Q),t(N1.

T(B)T(A)

The linearity of t follows from that of T on env(G). Consequently, t is a representation of the Lie algebra g of G. Thus, every representation T of G leads to a representation t of the Lie algebra g of G. Unfortunately, the converse is false. A representation t of the Lie algebra g need not supply a representation T of G. Nevertheless, the introduction of an associative algebra U(g) such that g can be identified with a subspace of U(g) with [x,y] i n g corresponding to xy - yx i n U ( g ) proves to be fundamental. Warning! This algebra is not the algebra env(G) introduced above.

(4.50) DEFINITION. Let g be a Lie algebra over the real (complex) numbers. A pair { U(g), i} where U(g)is an associative algebra over the real (complex) numbers and i is an injection of g into U ( g ) is called a universalenveloping nl~qcbraof g if: Given any associative algebra A and a map f:g + A which is linear and such that f ( [ x ,y]) = [f(x),f(y)] for x, y E g, there exists a unique homomorphism /7 : U ( g ) -+ A such t h a t f = hi. The map /z is a homomorphism of the associative algebra U(g) into the associative algebra A. The universal enveloping algebra U(g) proves to be unique up to isomorphism. The construction of a satisfactory model and the establishment of all the required properties is a sophisticated piece of mathematics which we do not attempt. See Jacobson (1962, Chap. V). The associative algebra U ( g ) has two fundamental properties from the standpoint of representation theory. The first of these is the following theorem. (4.51) THEOREM. Let g be a real (complex) Lie algebra and V be an rndimensional vector space over the real (complex) numbers. There is a natural one-to-one correspondence between the set of all representations of g on V and the set of all representations of U ( g ) on V where { U ( g ) ,i> is the universal enveloping algebra of g. If t : g+gI,(V) is the representation of g and T : U(g) Hom(V, V) is the corresponding representation of U ( g ) , then --f

t(x) = T(i(x)),

x

E

g.

The second fundamental property is that the representation theory of U ( g ) can be uwrked out in detail for important cases.

299

4. The Representation Theory of Certain Matrix Groups

We must introduce additional terminology in order to discuss further results. Our remarks are restricted to a Lie algebra g over the field F of either the real or complex numbers. Most of the statements are true for Lie algebras over fields of characteristic zero. Let X and Y be subsets of g. The symbol [X, Y] denotes the linear span of all elements of the form [x,y], x E X, y E Y. (4.52) DEFINITION. A subspace f of the Lie algebra g is a subalgebra of g if and only if [f, €1 c f . This asserts that the subspace f is closed under the Lie product.

A subspace b of the Lie algebra g is called an ideal of g (4.53) DEFINITION. if and only if [g, b] c b. The 0-subspace is always an ideal of g. One can show by means of the Jacobi identity that if b an ideal of g, then [b, g] = [g, b] is an ideal of g contained in 6. Furthermore, [b, 6'1 is an ideal whenever b and b' are ideals. (4.54) DEFINITION. The sequence of ideals, defined recursively by g1 = [g, g] and g'+' = [g', g], 1 i, forms a descending chain (4.55) of ideals which is called the lower central series of g. (4.56) DEFINITION. A Lie algebra g (an ideal or subalgebra b of g) is called nilpotent if and only if the lower central series of g (of 6 ) terminates in the zero ideal after a finite number of steps. (4.57) DEFINITION. Let g be a Lie algebra. The sequence of ideals of g, defined recursively by g' = g(l) = [g, g] and g ( ' + l )= [g"), g")], 1 5 i, forms a descending chain (4.58)

g(l)

...

g(n)

.. .

called the derived series of g .

A Lie algebra g (an ideal or subalgebra b of g) is called (4.59) DEFINITION. solvable if and only if the derived series of g (of b) terminates in the zero ideal after a finite number of steps. Every Lie algebra g contains a maximal solvable ideal n (4.60) DEFINITION. called the radical of g. A Lie algebra is called semisimple if and only if it has radical (0). Let g be a Lie algebra with no proper ideals for which g' = [g, g], the derivedalgebra of g, is not (0). Then g is a simple Lie algebra. The structure theory of finite-dimensional simple and semisimple Lie algebras over the complex field is known in great detail. The term Lie algebra, without additional qualifications, denotes a simple or semisimple Lie algebra over the complex or real field in the sequel. A Lie algebra of linear transformations is a subspace

300

4. Representation Theory of Special Groups

S of Hom(V, V) such that A , B E S implies that A B - BA E S . All of the Lie algebras of the standard groups are Lie algebras of linear transformations.

(4.61) DEFINITION. Let g be a Lie algebra of linear transformations acting on an m-dimensional vector space V over F. A linear mapping LY: g --t F is called a weight of g with respect to V if there exists a non zero vector v E V such that ( A - M(A)I)”‘A’V = 0 for some integer m(A), depending on A , for every A ~ gThe . set of all such vectors (including zero) for which this condition is satisfied form a subspace V, of V called the weight space of g corresponding to the weight a. (4.62) THEOREM. Let g be a nilpotent Lie algebra of linear transformations acting on the m-dimensional complex space V. Then g has only a finite number of weights with respect to V. Each weight space W of V is invariant under the action of g. Furthermore, V is the direct sum of the weight spaces of g. In addition, let

v =v,@ . . . @ V ,

be a decomposition of V into subspaces V i , 1 i i 5 k , such that each V i is invariant under the action of g. Suppose also that (i) the restriction of any A E g to Vi is a linear transformation with a single characteristic root a,(A) (not necessarily of multiplicity one) ; (ii) for i different from j , there exists B E g such that ai(B) # aj(B). Then the mappings a i : g + K are the weights of g with respect to V and the spaces Vi are the corresponding weight spaces. (4.63) REMARK. The methods of study of Lie algebras somewhat parallel those of the study of associative algebras. One represents a Lie algebra g on itself, so to speak. For any x E g, let ad x denote the element of Hom,(g, g) defined by ad x(m)

=

[x, m],

m Eg

It follows immediately from the definition that ad x is a linear transformation on g which is read “add ex.” The map ad: g + Horn&, g) is a representation, called the adjoint representation, of the Lie algebra g whose representation space is g. Note that the equations ad[x. yI@)

=

[[x, PI, ml = [x, [Y,m11 + [Y, [x,mll x(ad y(m)) - ad y(ad x(m)) = (ad x ad y - ad y ad x)(m)

= ad

follow from anticommutativity and the Jacobi identity. The result shows that the map ad preserves the Lie product. The remainder of the argument that

4. The Representation Theory of Certain Matrix Groups

301

ad is a Lie algebra homomorphism is easy to supply. The adjoint representation plays the same crucial role in the study of semisimple Lie algebras that the left regular representation plays in the study of semisimple associative algebras. A Lie algebra Ij is said to be abelian if and only if [lj, Ij] = (0). Every semisimple Lie algebra g over the complex numbers has associated with it a family of very special abelian subalgebras which satisfy the followingdefnition. (4.64) DEFINITION. A subalgebra Ij of the semisimple Lie algebra 9 is a Cartan subalgebra of g if and only if: (i) The subalgebra Ij is abelian, but is not properly contained in any abelian subalgebra of g, that is, Ij is a maximal, abelian subalgebra of g. (ii) For each element x E Ij, the linear transformation ad x, regarded as a linear transformation on g, is semisimple. This means that the Lie algebra g decomposes into invariant subspaces which are eigenspaces of ad x. This semisimplicity of the transformation ad x is at the basis of the analysis of semisimple Lie algebras over the complex numbers. The classification of the finite-dimensional semisimple Lie algebras over K is based on an analysis made possible by Theorem (4.62). Select a Cartan subalgebra Ij of g. Then g decomposes as the direct sum of weight spaces of ad I). (4.65)

g = Ij 0 w,, 0.. .0 wak.

The Cartan subalgebra Ij itself is the weight space corresponding to the zero weight of ad Ij. If one takes {Ij,, . . . , Ij,) to be a complex basis of Ij, then the dimension r is normally called the rank of the Lie algebra g. There may exist nonisomorphic semisimple Lie algebras of the same rank. Each weight space Wmicorresponding to a nonzero weight aiproves to be one-dimensional over K. Furthermore, if tli is a weight, then - a i is a weight. However, tli, - x i , and 0 = Oa, are the only multiples of ai which are weights. Thus one can select a family of vectors {e,,) corresponding to the nonzero weights such that each Wuiis spanned by e a i . There is a great deal of arbitrariness in all of these choices. Consequently, to find a standard description of the Lie algebra g, all of the choices must be made in a very special way. The full treatment of the classification problem for a semisimple Lie algebra g over the complex field is an elegant piece of linear algebra beyond the scope of this book. A change of terminology is made when discussing the decomposition of g under the action of the niepotent Lie algebra ad Ij. The weights arising are called roots of g and their weight spaces called rootspaces. A special class of roots (or weights), known as simple roots, is needed in the analysis. Their definition requires the introduction of an ordering in a real subspace Ij,* of the dual space Ij* of Ij. One denotes by IjR* the set of all r e d linear

302

4. Representation Theory of Special Groups

combinations of the roots of g. The real linear space selecting any basis {I.,, . . . , 2s} for it. Then an element =

bR* is

ordered by

r, J . ~+ . . . + <,$a,

is said to be pasitirc or grcwtc’r //inn 0 if and only if its first nonzero coefficient ti is positive. The vector x is greater tlzan the vector y if and only if the e x - y is positive. A yositice roof a is one such that 0 < a. (4.66) DEFINIT~ON. A simple root is a positive root which is not the sum of two positive roots. Clearly one obtains different sets of simple roots for different choices of the basis ti.,, . . . . AT}. It turns out that any set {al,. . . , a,.} of simple roots is a real basis of IjR* and a complex basis of Ij*. Thus the number of vectors in any set of simple roots is equal to the rank of the semiGniple Lie algebra g. Every root a can be written TX

=nlctl

+ ... + n,a,

;IS an infcyrnl h c u i . c o ~ h i n a t i o nof the simple roots where the set (n,} of integers are either all nonnegative or nonpositive. When 0 < a, they are all nonnegative. When s( < 0, they are all nonpositive.

(4.67) DEFINITION. There exists a symmetric bilinear form ( , ) on every finite-dimensional Lie algebra g. This form is called the Killing form and is defined by

(x,y)

= tr(ad

x a d y),

x, y

E g.

It is a famous theorem of Cartan’s that a Lie algebra over a field of characteristic 0 is semisiinple if and only if its Killing form is nondegenerate.

The Killing form remains nondegenerate when restricted t o a Cartan subalgebra 11 of a semisimple Lie algebra $1. This means that given any root a in i)*, there exists a unique element h, in Ij such that .(X)

= (h,.

x

x),

E

1).

where ( , ) denotes the Killing form restricted t o 5. The correspondence r/ + h, enables one to define an inner product { . 1 on I?,*. Given roots a and [I i n detine { a , /I> = ( h a

>

ha).

where { ,I denotes the inner product on bR* and ( , ) denotes the Killing form on Ij. The space I),* is an r-dimensional real Euclidean space under the norm I/ / / induced by { , >. Given a set {ri} of simple roots of g. the Lie algebra is completely characterized by nieans of a canonical selection of elements {hi,e i ,fi} for each

303

4. The Representation Theory of Certain Matrix Groups

simple root, 1 I i 5 r, together with a certain r x r integral matrix called a Cartan matrix of g relative to the Cartan subalgebra 5. These matters are efficiently described by means of Dynkin diagrams. We will present some examples of Dynkin diagrams in the sequel. Given a root a , from the set {g1, . . . , ar} of simple roots, there exists a unique element hai E 5 such that .i(x)

x 9.

= (x, hai),

Select any root vector ea,corresponding to for x

E

5. Since

E

such that

[x,euil= ai(x)eai= (x, hui)ea,, - x i is also a root, there exists a root vector e P a isuch that [x, e - J = -ai(x)e-ai

for x

CI,

=

-(x, hui)e-u,

b. It turns out that [eai,e-,J

=

e-,,)haZ

and that e-ui can be selected such that (eai,e-ui)= 1. The elements eal and e-,i are unique only to scale factor, but this last condition is a partial normalization. One defines the set {hi, e , , fi} by (4.68)

hi = 2hui/{ai,a,},

e, = eai,

f,

= 2e-,,/{ai, .,}.

These vectors satisfy the following multiplication table. (4.68)’

[hi,ej] = Aijej, [hi, fj]

where the matrix { A i j } , A i j matrix.

=

= 2{r,,

- A . 1J. f .J ’

[e,, fj] = 6 IJ. . hJ .’

a j } / { a i ,a , } , 1 I i, j

< r,

is the Cartan

The elements of the set {el, . . . , e,} are called simple or elenirntary raising operators while those o f the set {fl, . , . , f,} are called simple or elementary lowering operators. (4.69) EXAMPLE. Up to isomorphism, there is only one simple Lie algebra g of rank one over the complex numbers. A Cartan subalgebra of g is spanned by all complex multiples of a vector h E 9. There are two root spaces W, and W-, corresponding to the nonzero roots a and -3. The set {h, e, f } corresponding to the simple root CI is a basis of g for which the multiplication is determined by (4.71‘). Jacobson (1962) refers to this algebra as the split threedimension simple Lie algebra. The adjective split is used to indicate that the characteristic values of all the elements of ad 1, are included in whatever field of characteristic 0 is being considered. The real Lie algebras so(3) and Sll(2) are real forms. see below, of the simple Lie algebra g of rank one over the complex numbers. To obtain the

304

4. Representation Theory of Special Groups

complex Lie algebra g from so(3). one merely takes all complex linear combinations of the set { A l , A , . A 3 } of matrices spanning 5-43). The physicists are accustomed to introducing a different set { I f j } , 1 < j _< 3, of generators as an intermediate step. These are matrices defined by the equations

1
Hj=iAj,

The commutators of these elements assume the form (4.70)

[HI. H,]

= iH,,

[ H , , H,]

= iH,,

[ H , , HI] = iH,

for this new basis. Raising and lowering operators are defined by H,

=HI

+ iH,.

H-

= HI

- iH,,

so that H , , H- , and H , give still a different basis for g in which the commutators assume the form [ H 3 , H + ]= H +

.

[Hj, H - ]

=

[ H +, H - ] = 2 H 3 .

-H-,

Finally. one obtains a cawnical basis for g by defining (4.71)

e=H+.

f=H-.

h=2H,

The commutators for the set {h, e, f } are given by (4.71')

[h, e] = 2e,

[h, f ] = -2f,

[e, f ] = h.

The three-dimensional simple Lie algebra g over the complex numbers and its representation theory play a central role in both the structure and the representation theory of semisimple Lie algebras. The principal theorem on the irreducible representations of g , see Jacobson (1962, p. SS), is the following. (4.72) THEOREM. Let g be a three-dimensional simple Lie algebra over the complex numbers. Then for each nonnegative integer rn there is, u p t o isomorphism, exactly one irreducible g-module V of dimension m + 1. The space V has a basis {v,. , . . . v,,,} such that the actions of the members of the set {h, e, f } are given by

(4.73)

hvj = (t72 - 2 j ) v j , fVj = v j r l , fv, = 0, ev, = 0,

Osjsm.

evj = j [ m - j + I]vj-,,

1 2 j < m.

O _ < j < m - 1,

(4.74) REMARK. Each such (m + I)-dimensional representation is characterized by the integer n7 which is the highest eigenvalue of h which occurs. The weight CI which assigns to h the eigenvalue m(h) = rn is called the highest weight

4. The Representation Theory of Certain Matrix Groups

305

of the representation. All complex finite-dimensional, irreducible representations of any semisimple Lie algebra are characterized by such a highest weight. It is perhaps worth noting that the family of representations described above is frequently seen in the physics literature, but in a slightly different normalization. According to Eq. (4.71), h is taken to be 2H,, where H , is the generator of the Cartan subalgebra generally used by physicists. Consequently, H 3 has for its highest eigenvalue some integral multiple of in the irreducible representation described by physicists. The interested reader can find the details discussed in Gel’fand and Sapiro (1952, pp. 223-232). We have observed that the Lie algebra su(2) of the Lie group SU(2),the universal covering group of SO(3), is isomorphic to so(3). The finite-dimensional, irreducible representations of eu(2) and SU(2) are in one-to-one correspondence. Thus Theorem (4.73) completely determines the finite-dimensional irreducible representations of SU(2). Furthermore, all irreducible representations of SU(2) are finite-dimensional. The odd-dimensional, irreducible representations of m ( 2 ) % 5 0 ( 3 ) give rise to ordinary, irreducible representations of SO(3). However, the even-dimensional, irreducible representations of 50(3) correspond t o what are called spinor represenfations of SO(3). These come from those irreducible representations of SU(2) with kernels K not containing the kernel N of the natural map of SU(2) onto its factor group SO(3), that is, t o the irreducible representations of SU(2) which do not map

+

l -: -:I1

onto the identity. The spinor representations of SO(3) play a significant role in physical applications. They are discussed at some length in Gel’fand and Sapiro (1952).

(4.75) REMARK. The relationship between the uses of the real and the complex numbers in Lie groups and Lie algebras is quite bewildering on the first encounter. The Lie algebras which appear in a natural way are real Lie algebras, but the ones which get analyzed are complex Lie algebras. Since one begins with Lie groups whose underlying space is either a subset of real Euclidean space or at least is locally like real Euclidean space, the Lie algebra of tangent vectors at the origin forms a real vector space. The situation arises since differential geometers frequently deal with real geometries. Unfortunately, the algebraic problems connected with the solution of polynomial equations make it easier to deal with complex Lie algebras and to classify the complex simple and semisimple ones rather than the real. Fortunately, one can go backwards and forwards among the complex and real Lie algebras. Given any real Lie algebra g of real dimension k with basis {x~,. . . , x,}, there exists a complex Lie algebra gc, called the complexijcation of g, of

306

4. Representation Theory of Special Groups

complex dimension k whose elements are all formal linear combinations x = c i x i ,ci E K . The operations are defined by

2

%(Ici xi) 1 ixcixi, c cixi + In i x i = 1 + =

[Ici xi 1 d j *

(Ci

Xi]

=

CI

E

K,

di)Xi

11 c,d,[x,, Xi].

This is, of course, 9 0K in an informal dress. Conversely, given a complex Lie algebra g of dimension k with basis {x,, . . . , xk), there exists a real Lie algebra gR of dimension 2k with basis {xi, _ . . .x , , i x , , ..., ix,) whose product is inherited from g itself. It sometimes proves possible to determine a real Lie subalgebra r of gR. such that g R assumes the form r -t ir. Such a real subalgebra r of gR is called a real form of g itself. Generally, a complex Lie algebra g may have several different forms. In the case of a complex, simple Lie algebra, among the real forms there is always precisely one called the cor?ipact r.ml,fi)rriz. The term is used because the compact real form is ;i real Lie algebra which is the Lie algebra of a compact Lie group. The classification problem for real Lie algebras can be thrown back into the classification problem for complex, but the task is a nontrivial one. Each simple real Lie algebra can be one of two mutually exclusive types: (i) simple complex Lie algebras regarded as real algebras, or (ii) real forms of simple complex Lie algebras. A good discussion of these matters can be found in Helgason (1962, see i n particular pp. 152-156), and a less formal one i n the work of Belinfante and Kolman (1972). Any finite-dimensional g-module V of a seniisimple Lie algebra g is also an 6-module for any Cartan subalgebra b of g. Therefore V decomposes as the direct sum of weight spaces of 5. Each weight I of b is an element of the space bR*, that is, each weight is a real linear combination of the roots of l). Consequently, there is an ordering of the finite set of weights of l) belonging to V. (4.76) D E I ' I N I T I O N . The largest among the weights of called the /zig/zrst w i g h t of on V.

l) belonging

to V is

Hence [I,* contains a set of highest weights of all of the irreducible gmodules. Given 0-modules U and V, their tensor product U 0V is also a 9-module. If u @ v E U @ V and x E g, then x(u Ov) = X U O V

(4.77)

+UOXV,

a rule which is a bit surprising. Let lL and A be the highest weights of the irreducible g-modules U and V with weight vectors u and v, respectively. Then one has X(U

8 V)

= I.(x)(u 0V)

+ A(x)(u @ V)= (L(X)+ A(x))(u 0 V)

4. The Representation Theory of Certain Matrix Groups

307

so that U @ V has A + A for a weight which proves to be the highest weight of U @ V. There is a standard process for finding an irreducible g-module X of U @ V with I + A as its highest weight. This irreducible submodule X is called the Cartan composition of U and V. Thus, if 1 and A are highest weights, then I + A is also a highest weight so that the set of bR* consisting of the highest weights of g is closed under addition. A highest weight I is said to be a basic highest weight of g if it is not the sum of two other highest weights. The number of basic highest weights of g is equal to the number of simple roots of g. If A = {A1, . . . , A,} is the set of basic highest weights, then the set {al, . . . , a,} of simple roots may be enumerated so that (4.78)

2{4, c r j } / { c r j , a j } = d i j .

This equation says that the Bravais lattice i?* generated by the basic highest weights is (up to scale factors) the reciprocal lattice of the Bravais lattice i? generated by the simple roots. The Bravais lattice 2*plays a fundamental role in the representation theory of g. Every weight A of any finite dimensional representation of g corresponds to a point of the lattice 2*. The highest weight A of an irreducible g-module V is an integral linear combination

A =nlAl

+ ... + n,I,

of the basic highest weights with nonnegative coefficients. Jacobson uses a slightly different terminology. We recall that each mi of the set (q,. . . , a,} of simple roots is associated with a triple {hi, e , , fi}. All of these triples together determine a set B = {hl, . . . ,h,} which is a basis of 9. A linear functional I of b* is said to be integral if &hi) is an integer for every hi E B. One has Ai(hj) = (hn, ,hj) = (hn,, 2h,

/{aj,~ j } = )

2{&, a j } / { u j ,aj) = 6 i j .

Thus the basic highest weights and all their integral linear combinations are integral. An integral function A is said t o be dominant if A(hi) 2 0, hi E B. The nonnegative integral linear combinations of the basic weights, and only they, are dominant. The fundamental theorem on $nite-dimensional irreducible modules of a semisimple Lie algebra over the complex numbers is given below. See Jacobson (1962, Chapt. VII).

(4.79) THEOREM. Let g be a semisimple Lie algebra over K . Let b be a Cartan subalgebra of g. There is a one-to-one correspondence between the dominant integral functionals on E, and the finite-dimensional irreducible modules V of g. Let A be the highest weight of ij which occurs in V. Then 1 is a dominant integral linear functional on 8. Conversely, given a dominant integral linear functional A on b, there exists a finite-dimensional irreducible g-module V, unique up t o isomorphism, with A as its highest weight.

308

4. Representation Theory of Special Groups

The highest weight A of an irreducible g-module V has a one-dimensional weight space. Let v be any weight vector of A. Then eiv = 0 for every simple raising operator of g. Such a vector is called an extreme vector. Given a nonzero extreme vector v in any finite-dimensional g-module V, there is a systematic method of using v to generate an irreducible g-submodule of V by means of the elementary lowering operators. The set 2l3 of all weights of lj occurring in any irreducible g-module V constitute a symmetrical set in the Bravais lattice i?* spanned by all integral linear combinations of the set A = {Al, . . . ,A,}. The set 2B is called the weight diagram of V. The weight diagram determines V up to isomorphism since the highest weight in \II! is sufficient for this purpose. There are a number of rules satisfied by the weight diagram 9 . 3 of V of which we mention two : (i) For any root CI of g, there exist nonnegative integers p and q such that if w E %3, then w + na E 1' 13 for any integer n with - p I n 5 q. This sequence of weights is sometimes called the cc-ladder through w. (ii) For any root CI of g and any weight w of %J3, the functional w' = w - 2{w, a}/(cc,C Y belongs }~ to !ID. Geometrically the weight w' is the reflection of w in the plane through the origin in f&* perpendicularto the direction of CY. The dimension of the weight space of w' is the same as that of the weight space of w. The group generated by all such reflections is called the Weylgroup. In summary, the weight diagram is invariant under the Weyl group. We turn to the consideration of some examples. The simple Lie algebras over K of rank two have been investigated vigorously from 1960-1970 by physicists. The papers of de Swart (1963) and of Behrends et al. (1 962), among others, contain rather detailed applications of the ideas of this section t o these particular algebras. There are three nonisomorphic simple Lie algebras over K of rank two. These are SL[, , 'B2,and G 2 in the notation introduced by Cartan. The simple Lie algebra 21, of rank m, m 2 1 , over K is characterized intuitively by the following facts: (i) the real Lie algebra 91mRis isomorphic t o the real Lie algebra $,,+l(K) of the Lie group SLm+l(K). (ii) the Lie algebra 91, itself is the complexification of its compact real form eii(m+ l), the real Lie algebra of the Lie group SU(m+ 1). The dimension of PI, over K is m(m + 2). The simple Lie algebra 212 of rank two is the complexification of 4 3 ) and has dimension eight over the complex numbers.

The simple Lie algebra 23, of rank m, m 2 2, over K is characterized intuitively by the following facts:

4. The Representation Theory of Certain Matrix Groups

309

(i) the real Lie algebra SrnRis isomorphic to the real Lie algebra so(2m + 1, K ) which is the Lie algebra of the Lie group SO,,+,(K) (complex orthogonal group). (ii) the Lie algebra Bmitself is the complexification of its compact real form so(2rn + 1, R ) which is the Lie algebra of the Lie group SO(2m + 1, R) (real orthogonal group). Thus 8,is the complexification of the real Lie algebra of the Lie group SO(5, R) which is the group of rotations of a fivedimensional, real Euclidean space. The complex dimension of Srnis m(2m 1) so that of 23, is ten.

-+

The Lie algebra 8,is called an exceptional Lie algebra. It does not arise as the Lie algebra of one of the classical groups. Its complex dimension is fourteen. There is a wealth of information on these kinds of things in Helgason (1962, Chapt. IX). The Dynkin diagrams of the three simple Lie algebras a,, 23, and 6, of rank two are (i) 212 (ii) 23,

- o---o -

D

(iii) 8,

The two circles in each of the three Dynkin diagrams show that a set

{a1, a,} of simple roots for each of these algebras contains two simple roots.

The single line in the diagram for 21u2implies that the angle between the simple roots a1 and c(, is 120". The double lines joining the circles of the Dynkin diagram for 23, imply that the angle between the simple roots a, and a, is 135". The triple lines in the case of (5, indicate that this angle is 150". Since each of the circles of (i) are of the same shade, the two simple roots of '? are I,of the same length. The fact that one is darker than the other in (ii) and (iii) means that one simple root is shorter than the other. For the case (ii) of 2 lines, the ratio of the lengths is J 2 : 1. For the case (iii) of 3 lines, the ratio of the lengths is J 3 : 1. There is a result of Brown (1964) stating that the sum of the squares of all the root vectors is equal to the rank of a semisimple Lie algebra. Since the dimension of %, is eight and its rank is two, the number of nonzero roots is six. They are all of the same length so that it follows from the result of Brown that the common length is J3/3. Similarly, one finds the length of the short root a1 of 8, is J6/6 and that of the long root a2 is ,/3/3. For 8,,one finds the short length is J3/6 and the long lj2. The three root diagrams are shown in Figs. (4.80)-(4.82).

310

4. Representation Theory of’Special Groups

(4.80)

(4.81)

(4.82)

- Y2

i

-r 2

t

3x1 + Y 2

L

21, i T 2 =

31, i

i,

2r,

=

i,

311

4. The Representation Theory of Certain Matrix Groups

The simple roots in each of these diagrams are labeled g1 and u 2 , and the basic weights are labeled 1, and 1,. Figure (4.80) is the root diagram of U , . The lengths and coordinates of the simple roots and basic weights are la1 I = J3/3,

121 I =

1/39

I a, I = $/3, \ A 2 1 = 1/3,

a1 = (J3/6,

lP),

4= (J%,

W),

a2 = (J5/6,

- 1/2).

= (J3/6,

- 116).

1,

Figure (4.81) is the root diagram of 23,. The lengths and coordinates of the simple roots and basic weights are

I a1I = J6/6, I 4 I = J3/6, I a2 I = J 3 / 3 , IA2 I = J6/6,

a1 = (,/3/6,

J3/6),

1, = (J3/6,0), x , = (0, - J3/3),

A2

= (J3/6,

- J3/6).

Figure (4.82) is the root diagram of 8,.The lengths and coordinates of the simple roots and basis weights are

( u l I = $/6,

al = (fiI12, 1/41,

1 1, 1

= &16,

Ic(2I

=

a2

I 1, 1

112,

=

112,

L2 = (J3/4,

4 = (J3/6,0), =(O,

- 1/21?

- 1/4).

We include a few weight diagrams of 41, to illustrate some of the general ideas. The reader is referred to the paper of de Swart (1963) for many others. (4.83) EXAMPLE. The weight diagrams for the basic (highest) weights L1 = (J3/6, 1/6) and ,I2 = (J3/6, - 1/6) are shown in Fig. (4.84),where 1, = wt =

312

4. Representation Theory of Special Groups

(\/3/6, 1/6). wZ = ( - J 3 / 6 . 1/6), w3 = (0, - 1/3), w,' = (0, 1/3), ,I2 = w2' = (,,h/6, - 1 /6), w3' = (-\/3/6, - 1/6). In Diagram (I). there are three examples of x-ladders. {wl. w3}, {wl,w2}, and {wz, w3}. The first of these is either an a1 or a - El-ladder. Considering it as an &,-ladder, one has p = 1 and q = 0 while the ladder consists of w, + n r , for - p = - 1 5 n 5 0 = q. The sides of the triangle are named with the positive root a of which the vertices constitute an a-ladder. All of the weights have a one-dimensional weight space so that each of the irreducible representations is three-dimensional. These two representations are famous in physics as the " quark " representations. We sketch two other weight diagrams to give the reader a feeling for the symmetry of them (Fig. 4.85). All weight diagrams for 91, tend toward triangular or i

(4.85)

vv

hexagonal shapes. In both cases, the horizontal and vertical axes pass through the center of the weight diagram. We wish to make a few remarks about the simple Lie algebra 913 of rank three over K which is the complexification of the real Lie algebra su(3). Since the Dynkin diagram of 213 is

a set { x i . x 2 , aj) of simple roots of the algebra contains three elements with a common length. The complex dimension of '$!I3 is 3(3 2) = 15 by our basic formula. I t follows that 213 has twelve nonzero roots which prove to be of equal length 4.The angle between a, and a2 is 120", that between ct2 and a3 is 120'. and that between a1 and a3 is 90". The nonzero root vectors are the twelve vectors which join the origin to the midpoints of the sides of a cube with center at the origin and orientation that of the coordinate axes. Geometrically, the picture is as shown in Fig. (4.86). There are many different sets of these twelve vectors which will serve as a set {x1,a 2 , a 3 } of simple roots. We have selected xl = ( p , 0, p ) , 2, = ( - p , p, 0), and a3 = (p, 0, - p ) to be specific. All of the positive roots are labeled in Fig. (4.86). The negative ones can easily be read from these. It is well known that the Bravais lattice spanned by these vectors is the face-centered cubic lattice. Therefore, the lattice

+

313

4. The Representation Theory of Certain Matrix Groups

(4.86)

spanned by the basic highest weights is the body-centered cubic lattice. We are speaking of the integral linear combinations of the simple roots and basic highest weights, of course. A summary of the data is:

I @I1 I 11 I I @21 I1, I

=

= $/4, =

1/29

= J2/4, =

113 I

112,

1/2,

= 4614,

@1 = (J5/4,0,

4 = (J5/8,

J2/4),

J%3,

42/81?

% = ( - J2/4, J2/4,0),

1,

= (0, J2/4,

@ 3 = (J2/4,0,

O), -J2/4),

d3 = (J2/8, J2/8, -J2/8).

The Lie algebras 81,. m 2 1, constitute one of the great families of classical Lie algebras. As such, there is a world of information available on them. We make a few brief remarks about an excellent summary of ltzykson and Nauenberg (1966). Here one can find detailed descriptions of certain Cartan subalgebras of \urnand the related elementary raising and lowering operators. In particular, there is a useful discussion of the relationships between the Lie algebra approach to the representations of BI, and SU(m+ 1) and the analysis of those of S U ( m + I ) determined by means of Young’s tableaux. We make two basic points: (i) Let I denote the family of irreducible representations of S U ( m + l ) which correspond to the irreducible representations of 91, arising from the basic highest weights of am. Then the set I of irreducible representations of

314

4. Representation Theory of Special Groups

SUfm+ 1) is the set of irreducible representations obtained from frames F with but a single column :

These tableaux determine the antisymmetric tensors of ranks 1, 2, . . . , m which are the representations of SU(m + 1) arising from the basic highest weights of %, . (ii) The representation and character theory of the symmetric groups can be used with considerable effectiveness in the analysis of the representations of SU(m+ 1). We recommend this paper to readers who have practical computations to make concerning the representations of SU(m + 1). PROBLEMS

1. List the ordered partitions of 2, 3, 4, and 6. 2. List the frames corresponding to the partitions of 2, 3, 4, and 6.

3. Write down all the tableaux which correspond to the frame ' . . 4.

(a) Write down all the tableaux which correspond to the frame ' . (b) Write out the action of (123) E S3 on each member of this list.

5.

Determine e(T) for each tableau T of Problem 4.

6 . Given

123 T, =45 6

and

125 T2 = 3 6 , 4

find a permutation s E S6 such that e(T,)s = se(T2).

7. Given the tableaux

show directly that e(T2)e(7',)= 0.

315

Problems

8. (a) Write down all the frames belonging to 6 . (b) Determine the number of elements in the conjugacy class of S6 belonging to each of the frames. (c) Write down a typical permutation from each class. 9. Carry out the details of Lemma (1.19) for the group S, and some tableau T corresponding to (2, l}. 10. Find the dimension of the irreducible representation of S8 corresponding

to the frame

.... : . * ’

11. (a) Find the dimension of the irreducible representation of S7 corre-

....

sponding to :* . (b) Write out the set of standard tableaux corresponding to this frame. 12. (a) Find the dimension of the irreducible representation of Slo

.....

corresponding to the frame : ‘ . (b) Determine the lengths of the principal hooks in the hook diagram. (c) Determine the standard box with entries the principal hook lengths. (d) Find the value of the character of the corresponding irreducible representation for the class K with cycle structure 2’ 3’.

13. Find the value of the character of the irreducible representation of S , corresponding to the partition (4, 3, 1, l} for the class K with cycle structure 1 2 3,. 14. Find the matrix of the transposition (23) corresponding to Young’s rational seminormal form for the irreducible representation of S5 determined by the partition (3, 2). 15.

Let

=)It 4

=I1

112 1/2 and e2 1/2 1,211 denote idempotent generators of the minimal left ideals J, and J, of the

algebra K , of 2 x 2 complex matrices. (a) Show that K2 = J, @ J, even though the product e1e2 is not zero. (b) Find a decomposition of the identity 1 of K2 as the sum f, fi of orthogonal idempotents which are generators of J1 and J, , respectively.

+

16. Consider the irreducible representation T of S , determined by the frame (a) (b) (c) (d)

...

.

Find the dimension of T. Write down the list of standard tableaux of the frame. Determine the set of permutations pi,li in this case. Work out the matrix of the transposition (23).

316

4. Representation Theory of Speciul Groups

17. (a) Write out the Cayley table of the group S, . Take the elements of A , as the first twelve elements. (b) Compute the irreducible representations of S, by the methods of Chapter 3. (c) Check your answers by the methods of Chapter 4. 18. Determine a set of orthogonal idempotents for the decomposition of the group algebra of S, from first principals without using the techniques of Young outlined in the text.

19. Use the ideas of this chapter to give a general description of the irreducible representations of S, without going into details. Let M be a left A-module of the algebra A. Prove that the dual space M* is a right A-module according t o Definition (2.12). 20.

21. Let A be a symmetric algebra and M be an A-module. Show that D = Hom,(M, M) is a complex algebra. 22. Complete the argument thatf [ x [ y is an element of D = Hom,(M, M).

23. Consider the action of S, on M = V 0 V 0 V, where {vl, .. . ,vs} is a basis B of V. Let s denote the permutation (132). (a) Find the action of s on the vector m = (vl 0 v2 8 v,) + 2(v2 0 v3 0 v4) + 3(v, 0v4 0VS). (b) Write out the specific relations between the coefficients of m and sm in this particular case. Let M = V 0V 0V, where V has a basis {vl, vJ. Let s = (123) belong to S, and let g E GL(V) have the matrix

24.

1; 4

with respect to the given basis. (a) Verify directly that s(mg) = (sm)g, where m is the basis tensor v1 Ov, 0 v 1 . (b) Carry out the same direct verification for other extended basis elements for M. Let M = V @ V 0V 0V, where {vl, v2 , v3} is a basis of V. (a) Determine the number of equivalence classes of the relation R associated with the space of symmetric tensors W of M. (b) Select a standard representative element from each set.

25.

Problems

317

26. Let M = U @ U @ U @ U, where {ul, uz} is a basis of U. Let T =

123 4

and let W be the space of tensors symmetric under the rows of T. (a) Sort out the index diagrams into equivalence classes. (b) Select a row-ordered index diagram J R from each class.

27. Assuming the space U of Problem 26 has a basis {q,. . .,u,}, determine the set of standard index diagrams of T in the case where the set {jl,. . . ,j4} consists of the integers 1 through 4.

28. Let 1234 T = 567 89

be the canonical tableau of the frame diagram

.... :: . Denote by D the standard index *

1234 567 89 (a) Find the images qD where q runs through the set (15), (26), (37), (26)(37) and (269), each contained in the column group Q(T). (b) Row-order the resulting index diagrams of (a) and observe that D < p f q D whenever the new index diagram is standard.

29. Use row transformations to transform the column-ordered diagram E presented below into a standard index diagram. 6 3 12 8 14 11 19 13 22 18 24 20

1 2 523 4 921 7 16 10 17 15

30. Let M denote the GL(V)-module V @ V @ V @ V where V has dimension three. (a) Determine the dimension d of the canonical GL(V)-module N(F, V), where F is the frame :. ' . (b) Write down the standard index diagrams for F and compare their number with the dimension determined in part (a).

318

4. Representation Theory of Special Groups

31. Let M be the GL(V)-module of Problem 30. (a) Find the dimensions of the irreducible representations of S, determined by the frames:

(b) Find the dimensions of the standard canonical GL(V)-modules N(F,. V), W,, V). N ( F , , V), NF4,V) and N ( F 5 , V). (c) Write down the symbolic decomposition of M into irreducible S,-modules. (d) Write down the symbolic decomposition of M into irreducible GL(V)-modules. (e) Show that the dimensions check for parts (c) and (d).