Chapter 2 The Representation Theory of Finite Groups

Chapter 2

The Representution The09 of Finite Groaps

A fundamental tool in abstract algebra is the analysis of an abstract algebraic object A by means of a homomorphism h of A into a more concrete algebraic object B . The term representation is applied in the case where h is a homomorphism of A into the algebra of all linear transformations on a vector space V. The theory of group representations concerns itself with the classification of the homomorphisms of an abstract group G into groups of linear transformations or matrices. This chapter develops the representation theory of finite groups over finite-dimensional vector spaces. The field K of the vector spaces is frequently (not always) taken to be the complex numbers. The mathematical treatment is at a level between that of representation theory for physicists as presented by Wigner (1959) and that of representation theory for mathematical specialists such as given by Curtis and Reiner (1962). Since it is hoped that this book will serve as a point of departure for rhose who plan to read Curtis and Reiner, an effort has been made to conform to the definitions and terminology introduced by these authors. However, the basic analysis is done in terms of the group aIgebra of a finite group rather than in terms of a semisimple algebra in order to present the material at a somewhat lower level of mathematical sophistication. In the first section of this chapter we introduce the basic definitions of the theory o f group representations and give examples to illustrate them. The representation theory of finite abelian groups is worked out completely. It is 64

1. Basic Concepts and Defnitions

65

shown that every finite-dimensional representation of a finite group over the field K of complex numbers is completely reducible. In the second section we introduce the concept of the group algebra KG of a finite group G over the complex numbers K along with examples and fundamental definitions. The decomposition of the group algebra KG into the direct sum of minimal ideals is obtained. In the third section we develop the concepts and terminology of the theory of semisimple algebras in terms of the group algebra KG. A more detailed examination is made of the decomposition of the group algebra KG into the direct sums of minimal left- or minimal two-sided ideals. Section 4 contains a full account of the Pierce decomposition of a simple algebra in terms of a minimal two-sided ideal in KG. The number of classes of equivalent irreducible representations of a finite group G is shown to be equal to the number of distinct classes of conjugate elements in G. In the fifth and last section we define the character of a representation of the group G and establish the theorem that two representations of G are equivalent if and only if they have the same characters. The more important facts are established about character tables and illustrations provided in the case of certain groups. We summarize some of our notational conventions for the convenience of the reader. Most of the groups to be discussed are finite and commonly denoted by G and G'. Exceptions to this occur primarily in the cases of the full linear group GL(V) o f all invertible linear transformations of the vector space V and the full matrix group GL,(K) of invertible n x n matrices over the field K . Elements of groups are frequently denoted by either g and g' but many exceptions occur. Subgroups are sometimes denoted by H and K so that the symbol K is used both for subgroups and for fields. Vector spaces are denoted by U, V, and W. Elements of these spaces by boldfaced letters such as x, y, and z. Elements of fields are frequently denoted by lower case Greek letters such as a, j?, and y. Homomorphisms are denoted by I? and f as well as by T i n the case of linear representations. Linear transformations are also denoted by T . 1. BASIC CONCEPTS AND DEFINITIONS IN THE REPRESENTATION THEORY OF FINITE GROUPS

Let 6denote a subset of Hom,(V, V), the set of all endomorphisms of a finite-dimensional vector space over K. To say that 6 is reducible means that V contains a nontrivial subspace U which is a n invariant subspace of every linear transformation T belonging to 6. 6 is irreducible means that no nontrivial subspace U of V is an invariant subspace of each element T of 6. 6 is decomposable means there exist nontrivial, invariant subspaces U and W of V such that V is the direct sum U 0 V. 6is indecomposable means that

66

2. The Representation Theory of Finite Groups

there exists no nontrivial, invariant subspace U with a complementary, nontrivial invariant subspace W. G is completely reducible means that whenever U is a nontrivial invariant subspace of each T in 6, then there exists a second, nontrivial invariant subspace W such that V is the direct sum U 0W. The preceding definitions are meaningful for any family 6 of linear transformations belonging t o Hom,(V, V). They are used mainly in the case of a homomorphism T of a finite group G into GL(V). Here 6 is taken to be the Im T , the set of all elements of GL(V) of the form T(g) with g in G. Note that these definitions apply equally well t o the case of a family 6 of matrices contained in GL,(K). Here the vector space V is taken to be the coordinate space K". We now turn t o the most important definition in this chapter. Let G be a finite group and let V be a vector space over the complex numbers K. (1.1) DEFINITION. A linear representation of G with representation space V is a homomorphism T with domain the group G and range the full linear group GL(V).Two representations T and T' of G with representation spaces V and V' are said t o be equivalent if there exists an isomorphism Iz of V onto V' such that

T'(g)/z = /zT(g),

g

E

G.

The dimension (V : K ) of V over the field K of complex numbers is called the degree of T and denoted by deg T. The Im T is a subgroup G' of GL(V) which is contained in Hom,(V, V). The representation T is said t o be reducible, irreducible, decomposable, indecomposable, o r corripletely reducible if and only if G' is a reducible, irreducible, decomposable, indecomposable, o r completely reducible set of linear transformations in Hom,(V, V).

(1.2) DEFINITION. A matrix representation of G of degree n is a homomorphism T of the group G into the full matrix group GL,(K). The matrix representations T and T' are equizalent if they both have the same degree n and there exists a fixed matrix A in GL,(K) such that T'(g) = AT(g)A-',

LJ E

G.

The concepts of reducible. irreducible, decomposable, indecomposable, and completely reducible are immediately applicable to matrix representations as well as linear representations. The concept of equivalence for representations is an equivalence relation 91 on the set 5 of all linear o r all matrix representations. This statement means that (1) T i s equivalent to T for

67

1. Basic Concepts and Definitions

every representation T ; (2) T is equivalent t o T' if and only if T'is equivalent to T ; and (3) T is equivalent to T' and T' is equivalent to T" imply that T is equivalent to T". Thereby, the classification problem for representations is largely reduced to the characterization of all distinct equivalence classes of representations of a given group rather than to the determination of all of its individual representations. It evolves that it is sufficient to characterize the distinct classes of equivalent irreducible complex representations of which there are only a finite number in the case of a finite group. Therefore, our principal efforts in the following pages are to determine methods of finding all of the distinct classes of equivalent complex irreducible representations of a given finite group G. Each representation T of a finite group G is a homomorphism of G into G', a subgroup of the full linear group GL(V)over the representation space V of T. The adjectives reducible, irreducible, decomposable, indecomposable, and completely reducibte are extended from the representation T to the representation space V so that one speaks, for example, of an irreducible representation space V meaning, of course, the representation space of an irreducible representation T . The following facts are true of a representation T, either linear or matrix, merely because T is a homomorphism of a group G into a group G':

and

T(1)

=I,

T w o results about linear transformations and homomorphisms are particularly useful in giving specific examples of representations of a finite group G. First, there exists a unique linear transformation T in Horn#, V) for any specific choice of the set {Tv,, . . . , Tv,} of images of a basis {v,, . . . v.} of V. Second, it is sufficient to show that a mapping T with domain a group G and range Hom,(V, V) has the following properties: (1.3)

(i) T(gg')v, = T(g)T(g')v, for g,g' in G, and vi any element of { V I , . . ., v,>, (ii) T ( l )is the identity 1, in Hom,(V, V),

in order to show that T is a representation of G. We give several examples of representations of groups before continuing with the general discussion.

68

2. The Representation Theory of Finite Groups

For our first example, let G denote the cyclic group C, of (1.4) EXAMPLE. order four whose Cayley table is as shown in Table (1.4').

(I .4')

CAYLEY TABLEOF C, 1234 11234 22341 33412 44123

Let V denote a four-dimensional vector space over K with a basis consisting of the set {vl, . . . , v4). For each i, 1 < i < 4, define T(i)by the formula T(i)vj= v i +j

,

where the subscript i +.j is to be computed from the Cayley table of C4.The fact that the correspondence i 4 T(i)is a representation of C, can be shown by verifying that conditions (i) and (ii) of (1.3) are satisfied. (i) T(i+j)v, = V c i + j ) + k = v i + ( j + k ) = 7'(WWk1 = CWT(j)lvk; (ii) T(l)v, = vL+, = vk so that T(1) =I,. The family of linear transformations {T(i)}can also be described by means of their matrices { M ( i ) } .These are =

1 0 0 0 O O O 0 0 1 0 ' 0 0 0 1

M(3) =

0 0 1 0 0 0 0 1 1 o o 0 ' 0 1 0 0

M(1)

0 1 M(2)= 0 0

0 0 1 0

0 0 0 1

1 0 0 0

0 1 0 0 w4)=

0 0 1 0

o o o

1 1 0 0 0

It is simple, although tedious, to verify from their matrices that the set {T(1). T(2), T(3),T(4)) of linear transformations satisfy the same multiplication table as the group C,. Again we find that the mapping T of C, into Hom,(V, V) is a linear representation of C., This linear representation T gives rise to a matrix representation M which makes the matrix M(i) correspond to each element i of G. The linear representation T is said to uflord the matrix representation M . More generally, if T is a linear representation of a group G with representation space V and the set {vl, . . . , v,} is a basis B of V. then each linear transformation T(g),g E G, has a matrix M ( g ) with respect to the basis B. The correspondence M which makes correspond to each g of

69

1. Basic Concepts and Definitions

G the matrix M ( g ) is a matrix representation of G which is called the matrix representation afforded by T .

We consider a second example of broad significance in both representation theory proper and in its application to other disciplines. (1.5) EXAMPLE. Recall that a transformation group G on a nonempty set S is a group, each element g of which induces a bijection on the set S, such that if g, g’ are elements of G and x belong to S, then (gg’)x = g(g’x).

In greater generality, a group G is said t o act as a transformation group on the nonempty set S if there exists a homomorphism h of G into the group P ( S ) of all permutations on the set S. Familiar examples of transformation groups are GL(V), the group of all invertible linear transformations on a vector space V, and U,, the group of all unitary transformations on an n-dimensional inner product space. We illustrate a standard procedure for determining a representation of a transformation group G acting on a nonempty set S. Let V(S) denote the vector space of all functions with domain S and range the field K of complex numbers. We recall that V(S) is a vector space over K with (i) [ f + g](x) defined to be f ( x ) + g ( x ) and (ii) [ufJ(x) defined to be uf(x). We define a linear transformation T(g) belonging to Hom,(V(S), V(S)) for each g in G and show that this correspondence T is a representation of G. I f f E V(S) and g E G, we define T ( g ) by its action on f. that is, T(df=fog-1. It can be shown that T ( g ) is a linear transformation on V(S). Let g,g’ be elements of G. Then T(gg’)f = f o (gg’)-I

so that (1.6) Furthermore,

so that (1.6‘)

=~

=f 3 [(g’)-l 0 g-

‘ 1 = [ S o ( g ’ ) - l ] 09- ’

m T ( g ’ ) f= l LWT(g’)lJ;

Th’)= T(g)T(g’). T(l)f=fo1-1

=fo

T(1) = 1”(S)

1 =f, ’

It follows from (1.6) and (1.6‘) that T is a homomorphism of G into GL(V(S)) and, consequently, is a representation of G. This argument is entirely independent of the number of elements in either G or S.

70

2. The Representation Theory of Finire Groups

A simple generalization of this example can be obtained by letting V(S) denote the vector space of all functions with domain S and range a vector space U over the field K of complex nur,ibers. The above arguments extend to this case with virtually no modification and demonstrate that there exists a representation T of G with representation space V(S) in this instance as well. To be more specific. let S consist of the letters {a, b, c} and V be a three-dimensional vector space with the set (va. v b . vc} as a basis. Let the symbol S 3 denote the symmetric group on the set {a, 6, c} and denote its elements by g1 = (a),g z = ( a b c), g 3 = ( a c b), g4 = (a b), gs = (a c), and .96 = ( b c). A functionfE V(S) can be defined by means of a triplet of vectors (u", u,,. u,) where f ( a ) = u,, f ( b ) = ub. and f ( c ) = u,. A basis of V(S) consists of the nine functions,f,, . . . . , , f 3 3 which are defined by f 1 1 = (vu

3

O, O),

fi2

= (vb

O).

f22

= ( O , vb

.f32

= (O,

f21

= (O.

vo

f3l

= (O.

O , v',).

3

1

O, O ) ,

0, vb).

O, O)?

f13

= (vc

>

f23

= (O.

vc >

f33

= (O,

0, Vc).

O).

>

O),

The linear transformation T(gl) corresponding to g1 is clearly the identity. We look at the image T(g,) under the foregoing scheme.

[T(g2)f1(a>= (J"9 2 -')(a> = f ( c > , [7-(92)fl(b)

= (.Po 9 2 - I P )

=f(a),

and [7-(g2>fl(c> =

Cf

0

9 2-

'>(c>= f ( b ) .

I t follows from these relations that 7-h72)f;* =.f*13

7-(g2)s,2 =.f2*.

T(g2)f21

=f31,

T(g2).f22 = f 3 2

[(.92).f31

=f;I.

T(g2).f32

-_

=flZ

7-(92)f,,

T(g2)f23 = f 3 3

7

T(g2)f33

1

=f23r 7

'f13.

The matrix M ( g , ) of T(q,) with respect to this basis arranged in dictionary order is given by 0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 1

1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0

71

1. Basic Concepts and Definitions

The interested reader will be able to work out the matrices of any other of the linear transformations T ( g ) for himself. Before turning from this example, we wish to observe that this last representation is reducible. If

is the inverse of any element g of G, then the effect of T(g)on any f of V(S) of the form (u,, u b , u,) is given by (1.7)

T(g)(ua >

ub

>

uc)

= (ua’

> ub’ > uc’>.

If the three components off are equal to each other, then (1.8)

T(s)f=.L

S f G.

Thus, for example, the subspace U of V(S) spanned by all complex multiples of (v,, v,, v,) is an invariant or reducing subspace of the representation. We return to the development of the general theory. Consider two families L and L‘ of linear transformations belonging to the two sets Hom,(V, V) and Hom,(W, W), respectively. Assume that each element of each of the two sets can be uniquely labeled with an element from the index set IT. Thus L consists of the set {T(n): 7c E rI} and L‘ of the set {T’(n): n E n}.Let A be an element of Hom,(V, W) such that (1.9)

T’(n)A = AT(7c),

7c E

rI.

(1.10) LEMMA. The family L is reduced by the Ker A and the family L‘ is reduced by the Im A . Proof. Let T(n)denote any element of L and v denote any element of the Ker A . Then AT(n)v = T’(7c)Avwhich is zero since v belongs to the Ker A . It follows that T(n)(Ker A ) is contained in the Ker A for each 71, and the Ker A is an invariant (reducing) subspace of L. On the other hand, if T’(n) is any element ofL’ and Ax is any element of the Im A , then T’(n)Ax = AT(7c)x which is an element of the Im A . Therefore T’(n)(Im A ) is contained in Im A for each n, and the Im A is a reducing subspace of L‘. (1.11) LEMMA.Let the family L defined above be an irreducible family, then either the Ker A must be the null space of V or the Ker A must coincide with V. Proof. Since L is reduced by the Ker A , it must be a trivial subspace of V, that is, the Ker A must be either the zero subspace or else coincide with V. (1.12) LEMMA.Let the family L‘ defined above be an irreducible family, then either Im A must be the zero space of W or the Im A must coincide with W. Proof. The proof is analogous to that of Lemma (1.1 1).

72

2. The Representation Theory of Finite Groups

(1.13) LEMMA (Scliur). Let V and W be vector spaces of dimension greater than zero over the field K of complex numbers. Let both L and L' be irreducible subsets of Hom,(V, V) and Hom,(W, W), respectively, the members of each set being labeled with elements from the set FI. If there exists a linear transformation A in Hom,(V, W) such that

(1.14)

T'(n)A = AT(n),

71

E

rI,

then either A is an isomorphism of V onto W or A is the null homomorphism. Proof. By Lemma (1.1 I), the Ker A must either be (i) the zero space of V or (ii) the entire space V. In case (i), the mapping A is a monomorphism and the Im A is not the zero space of W. Therefore, the Im A coincides with the space W by Lemma (1.12) and A is an epimorphism. Thus, in case (i) the mapping A is an isomorphism of V onto W. In case (ii), A is the null homomorphism of V Into W since the Ker A coincides with all of V. (1.15) LEMMA (Scliuv). Let L be an irreducible family {T(n): n E I'I} of linear transformations in Hom,(V, V) where V is a vector space. of positive dimension over the field K of complex numbers. If there exists a linear transformation A in Hom,(V, V) such that AT(7t) = T(n)A,

n E rI,

then A is a multiple of the identity automorphism 1, on V. K

Proof: Let U, be the nonzero eigenspace corresponding to some eigenvalue of A . Let u be any element of U, and T(n) be any element of L. From AT(n)u = T(n)Au = IcT(n)u,

it follows that T(n)u belongs to U, so that the irreducible family L has a nonzero invariant subspace U, which must coincide with the entire space V. Therefore, the linear transformation A is equal to ~ 1 " . (1.16) LEMMA.Let U be a subspace of the vector space V over the field K of complex numbers. Let the set (PI , . . . , P,) be a family of projections of Hom,(V. V), each of which has the subspace U for its range. Then the linear transformation

(1.17)

P = ( l / n ) ( P , +..'+P")

is also a projection of U. v

Proof. It is sufficient to show that P is an idempotent with range U. Let V, then PivE U for 1 I i I n, so that the range of P is contained in U.

E

73

1. Basic Concepts and Definitions

If u E U, then P i u = u for 1 i 2 n, so that Pu = u and U is contained in the range of P, that is, the range of P coincides with U. Furthermore, (1.18)

P2x = P(Px)

= Px,

x Ev,

so that P is idempotent. We are now able to establish the most fundamental theorem of the representation theory of finite groups. (1.19) THEOREM (Muschke). Let T be a representation of the finite group G of order n on the m-dimensional space V over the field K of complex numbers. Let U be a reducing subspace of T. Then U has a complementary reducing subspace W. Proof. Let R be any projection on the invariant subspace U and note that T(g,)RT(gi-') is also a projection Pi on U for every gi E G. It follows from Lemma (1.16) that p

-'>+

= (l/W(g,)RT(g,

*.

. + ~hl)RT( gf l- ')l

is a projection on U. Moreover, for each choice of g E G, T(g)PT(g-') is equal t o P since it is merely a rearrangement of the summands of P. Consequently,

m p = PT(g),

(1.20)

9 E G.

If I, denotes the identity linear transformation on V, then (1.21)

m",

= [I,

-PI

- PlT(S),

9 E G.

Furthermore, 1, - Pisaprojection and i fx belongstoitsrangeW then (1.22)

T(g)X

= T(g)[lv

= [I, - P]V,

-PIX = [l, - P]T(g)x E W

for every g E G. This implies that W is a reducing subspace of the representation T. However, every x belonging to V can be written x

= Px

+ (1,

- P)x

= x,

+ x2,

where x, belongs to U and x2 belongs to W. Also, if x is common to U and W, then x =P x

= (1,

- P)x = P(1, - P)x = 0,

so that V is the direct sum U 0 W of the reducing subspaces U and W. Thus W is a complementary subspace of U as was to be shown. LetQ be a linear transformation on a vector space V, which is the direct sum U, @ @ U, of invariant subspaces of D. Then Q defines a linear

74

2. The Representation Theory of Finite Groups

transformationDi on each invariant subspace U i , such that for each ui € U i , (1.23)

DiUi =nui.

Suppose that T is a linear representation of a finite group G on the representation space V which is the direct sum U, @ . . . @ U, of reducing subspaces. Since each Ui is an invariant subspace of T , T ( g )defines a linear transformation T i ( g ) of Hom,(U,, Ui) for each g in G. If g and g’ are in G, then for any ui in U i Ti(gg’)ui= T(gg’)Ui= [T(g>T(g’>lui == T(g)[T(g’)uil (1.24) = T(g)[Ti(g’)uil = Ti(g)[Ti(g’)uil = ITi(g)Ti(~’>lui.

Consequently, we see that (1.25) Let Ti denote the mapping from G into Hom,(Ui, Ui) whose value at g of G is the linear transformation Ti(g)introduced above. According to (1.25), Ti preserves the algebraic operations. Furthermore,

( I .26)

T,(l)u,= T ( l ) U i

= ui

for every u i E Ui where 1 denotes the identity of G. It follows from (1.25) and (1.26) that Ti is a representation of G. The representation T is said to be the direct sum of the set {TI, . . . , Tk} of representations. We denote this by

(1.27)

T = T l @..*@Tk.

It is important to realize that the sum indicated here is of a special nature and that T(g> = Tl(g) @ ’ ’ ’ @ T k ( g ) is not a conventional sum of linear transformations. Each of the representations Ti is the restriction of T to the corresponding subspace Ui and can be denoted by the symbol TI U i . The terminology previously introduced can be applied to the summands occurring in (1.27). Thus we speak of Ui as being reducible, irreducible, and so forth, accordingly as the representation Ti is reducible, irreducible, and so forth. respectively. Now apply Maschke’s theorem t o obtain the following theorem. (1.28) THEOREM. Let T be a linear representation of the finite group G on the n-dimensional representation space V over the field Kof complex numbers. Then either V is irreducible or else V is the direct sum U @U’ of invariant subspaces where U is irreducible, that is, the restriction of T t o U is irreducible.

75

1. Basic Concepts and Definitions

Proof. The theorem is valid for the case of an irreducible space V. If V is reducible, then V contains a nontrivial reducing subspace W, on which T defines a representation T, . Either W, is irreducible or else W, contains a nontrivial reducing subspace W, . In this fashion, determine a properly decreasing sequence of subspaces

w, 3 w, 3 f . . )

(1.29)

each of which is an invariant subspace of T. Since the representation space V is finite dimensional, the sequence (1.29) must terminate in an irreducible subspace W,. Denote W, by U and use Maschke's theorem to write

V=U@U',

(1.30) where U is irreducible.

(1.31) THEOREM. Let T be a linear representation of the finite group G on the n-dimensional representation space V over the field Kof complex numbers. Then either V is irreducible or else V is the direct sum U, @ * 0 U, of irreducible subspaces.

-

Proof. The theorem is valid for the case of an irreducible space V. If V is reducible, then Theorem (1.28) asserts that V is the direct sum U, @ U,', where U, is irreducible. Whenever U,' is reducible, it can be decomposed as the direct sum U, 0 U,', where U, is also irreducible. In this way, one arrives at an ascending chain

u, c u, o u , c u, O U , ou, c * . * , where the sums are direct with each summand irreducible. The finite dimensionality of the vector space V implies that this chain terminates with V. Hence there exists an integer k such that V is the direct sum

v = u , @ "'Ouk, where each summand is irreducible. Such a decomposition of V gives a corresponding decomposition of T as (1.32)

T = T, @ ... @ Tk

where each summand is the restriction of T to the irreducible subspace U i . Thus one obtains the following corollary. (1.33) COROLLARY. Let T be a linear representation of the finite group G on the n-dimensional representation space V over the field K of complex numbers. Then either T is irreducible or T is the direct sum of irreducible representations.

76

2. The Representation Theory of Finite Groups

(1.34) THEOREM. Let T be a linear representation of the finite group G on the n-dimensional representation space V over the field K of complex numbers. If V is the direct sum U, 0 0U, of irreducible subspaces and W is any reducing subspace o f V, then either W coincides with V or V is the direct sum of W and some of the U i . Proof. First, note that if W' and U j are reducing subspaces of a representation T, then the intersection W' n Uj is a reducing subspace of T, common to both. This observation implies that if Ui is irreducible, then either W' n Uj is the zero space or W' n Ui coincides with U j. Second, we see that if W and V coincide, the theorem holds; otherwise, there exists some U i not contained in W. Relabel this subspace Uj, and note that the sum

w, = w OU;,

(1.35)

is direct since W n U j , consists o f the zero vector alone. Either W, coincides with V or else there exists a second irreducible subspace, say Ui, , such that W, n Uj, i s the zero space and the sum

w, = w, 0ui,

is direct. By repetition of the argument, one produces an ascending chain

w c w, c w, c

(1.36)

*

of subspaces in the finite-dimensional space V. It follows that (1.36) must terminate with some space W, which coincides with V From the nature of the construction, it follows that

v =w@ui,0.*.0Ui,,

(1.37) as was to be shown

Let T be a linear representation o f a finite group G with representation space V which has two direct sum decompositions, U @ W and U @ W'. A theorem from the general theory of vector spaces asserts that W and W' are each isomorphic to the factor space VjU and hence are isomorphic to each other. This theorem can be extended to the more general case in which U, W, and W' are invariant subspaces of the linear representation T. The conclusion now becomes that the representation defined by T on W is equivalent to the one defined by T o n W'. The argument in the case of a linear representation T is very similar t o the argument for vector spaces and necessitates the introduction of a factor representation. We recall that if 6 is an element of Hom,(V, V) with U a nontrivial invariant subspace of 6, then 6induces a linear transformation 6' on the factor space VjU. Consequently, each T ( g ) defines such a T'(g) by (1.38)

T'(g)(x

+ U) = T(g)x + u,

x

+ u E vju

77

1. Basic Concepts and Definitions

f o r g E G. Thus one can use the representation T of G to define a representation T' of G with representation space V/U. For each g of G, let T'(g) denote the linear transformation on the factor space V/U that is defined in Eq. (1.38). Note that if g, g' belong to G and x + U is any element of V/U, then T'(gg')(x+ U)

= T(gg')X

+ U = T(g)T(g')x+ U = T'(g)[T(g')x + U] = T'(g)[T'(g')(x+ U)l

= [T'~g)T'(g')l(x

+ U),

so that (9

T'(g9') = T'(dT'(9').

Furthermore,

T'(l)(x + U)

= T(l)X

+ U = x + U,

which implies that (ii) T'(1) = I",", Consequently, T' is a representation of G with representation space VjU. The representation defined above is called the factor (1.39) DEFINITION. representation of T o n the factor space V/U. (1.40) THEOREM. Let T be a linear representation of the finite group G on the vector space V over the field K of complex numbers. Suppose that V is the direct sum U @ W of the reducing subspaces U and W. Then the factor representation T' of G on V/U is equivalent to the representation T"of G on the reducing subspace W. Proof. Let v be the natural homomorphism of V onto V/U and observe that Y maps x of V into x U of V/U. Denote by A the isomorphism of W onto V/U which is the restriction of v to the subspace W. Then

+

(1.41)

ATn(g)W

= AT(g)w = T(g)w

+ U = T'(g)(w + U)

= T'(g)Aw,

w

E

W , g E G,

so that

(1.42)

AT"(g) = T'(g)A,

9 E G.

Thus T and T' are equivalent representations of G.

If T' and T" are two equivalent representations of a group G with representation spaces V' and V" respectively, we sometimes say that the two representation spaces V' and V" are equivalent, written V' z V".

78

2. The Representation Theory of Finite Groups

( I .43) COROLLARY. Let T be a linear representation of the finite group C on the vector space V over the field K of complex numbers. Suppose that V can be decomposed in two distinct ways. U 0 W, and U 0 W,, into the direct sum of reducing subspaces, with the first summand common to both decompositions. Then the second summands are equivalent. In symbols, if (1.44)

V=UQW, =UQW,,

then W, and W2 are equivalent, that is, the representations Tl and T2 defined by T o n the reducing subspaces W, and W, are equivalent.

ProoJ: The representations T , and T2 are each equivalent to the representation T' on the factor space VjU and, consequently, are equivalent to each other. We are now prepared to derive the fundamental theorem concerning the reduction of representation spaces into irreducible subspaces by means of the results obtained in (1.40) and 1.43). (1.45) THEOREM. Let T be a linear representation of the finite group G on the vector space V over the field K of complex numbers. Let U and W be equivalent subspaces of V. Suppose that U has the decomposition (1.46)

U = U l Q . * .QU,,

while W has the decomposition (1.47)

W

= W, @ . . .

QW,

into irreducible subspaces of T . Then (i) the number of summands in (1.46) and (1.47) are the same and (ii) there exists some permutation { j , , . . . ,j,} of { I , . . . . k ] such that the subspaces U i and Wj, are equivalent.

Proof. The proof is by induction on the integer k . If k is 1, then U is irreducible so that t must also be 1. Clearly U, and W, are equivalent in this case. Assume, for purposes of induction, that the theorem is valid for 1 Ik I m. Let U and W have the two decompositions (1.48)

U=U,

and

( I .49)

W

=

@~..@u,@u,+l w, Q . . . o w , ,

respectively. Since U and W are equivalent, there exists an isomorphism A of U onto W such that

79

1. Basic Concepts and Definitions

Let W,' be the image of U, under A . Then W,' is an irreducible subspace of W equivalent to U,. Theorem (1.34) implies that W is the direct sum

w = W,' 0wi, 0. . . 0W & . It follows that

u* = U 2 0 . - . @ U , + ,zu/u, M

W/W,'w w *

= w i l@ . . . O W i r .

By the induction hypothesis, there are rn summands in W*, each of which is equivalent to one of those occurring in the direct sum decomposition of U*. On the other hand,

u, w u/u*w w / w * ,

so that W* contains every summand of W except one. say W". which must be equivalent to U,. This completes the induction and establishes the theorem.

(1.50) THEOREM. Let T be an irreducible linear representation of a finite abelian group G on the representation space V over the field K of complex numbers. Then V is one-dimensional. ProoJ: Let T(go) correspond to any go of G. Then

It follows by Lemma (1.15) that T(go) is a multiplef(g,)lv of the identity on V. Any nontrivial subspace U of V reduces T. Thus V has no nontrivial subspaces, that is, V is one-dimensional. (1.51) THEOREM. Let Ckbe a cyclic group of order k. Then there is a one to one correspondence between the family { T } of irreducible representations of C, and the set (5) of kth roots of unity.

Proof. Let T be any irreducible representation of ck and a any generator o f C,. The representation space V of any irreducible representation T of G has

a basis {v} consisting of a single vector so that the image T(a) can be taken to be the linear transformation such that T(a)v = cv.

It follows that

v

=

I V = I,v

= T(1)v = T(ak)v = [T(U>],V= l k v ,

so that 5 is a kth root of unity. The kth root of unity icompletely determines the linear representation T. Thus, given a particular generator a of the group G, there corresponds to each irreducible linear representation T a unique kth root of unity i.

80

2. The Representation Theory of Finite Groups .

If j > j’ and a j = aj’, then a’-.’

.I

= 1,which

j =j’

(1.52)

implies that k divides j - j ‘ or

+ mk,

which means that [j

(1.53) whenever

=ij’+mk = Y

1;

5 is a kth root of unity.

The correspondence which pairs with each element a/ of G the complex number ij is a well defined mapping from G to the field K of complex numbers, since aj = ai’ implies that 5 j = [j’ according to (1.52) and (1.53). Thus we can define for each g = aj of G a linear transformation T(g) on the one-dimensional space V. Let (v} be a basis of V and define T(g) by T(g)v

= T(aj)v =

pv.

The correspondence T is a representation of G with representation space V. This follows from T(gg’)v = T(ajuj’)v = T(ajfj‘)v = ij+ j‘v = (iJ[j’)v = [j((j‘v) that is, and so that

= T(ai”(aj‘)vI

= %d[T(g’)vI = [n)T(g’)lv,

T(1)v = T(ak)v = Ikv = lv

= l,v,

T(1) = l v .

(ii)

Thus the irreducible representations of the cyclic group correspondence with the set of kth roots of unity.

c k

are in one-to-one

It is a well-known theorem that every finite abelian group G can be written as the direct sum (1.54)

G

= (ml) @

*

. . @(m,)

with each of the cyclic summands (mi) having order c ( ~such that ailcli+l, 1 5 i 5 r - 1 . Every irreducible representation T gives rise to a family { T I ,. . . , T,} of irreducible representations of the subgroups (mi), 1 5 i I r. Each representation Tiis the restriction of T to (mi),that is, (1.55)

Ti = TI (mi),

15 i I r.

1. Basic Concepts and Definitions

81

By Theorem (1.51), if the representation space V has a basis {v} consisting of the single vector v, then each of the representations Ti is characterized by a complex number C i , an aith root of unity, such that (1.56)

Ti(mi)v= Civ,

15 iI r.

Thus the value of the representation T for any element (1.57)

g =j,m,

of G,

+ +j r m , *

*

. Cj3v, is completely specified by the r-tuplet (Cl,. . . , [,) of a,th roots of unity. (1.58)

T(g)v

= (C1j1

*

Furthermore, if T‘is any representation of G with {v}‘ a basis of its representa tion space V‘ and (1.59)

T’(m,)v’= CiV’,

then T‘ and T a r e equivalent representations of G. Conversely, if (C1, ,. . , 5,) is an r-tuplet of complex numbers such that Ciai = 1 , 1 5 i 5 r, then the correspondence f such that (1.60)

f ( 9 ) = Cl” . . . [ A

where g is given by (1.57), is a well-defined mapping of G into the field K of complex numbers. This correspondence allows us to define a mapping T from G into Hom,(V, V) V a K-space with basis {v}. T ( g )is defined by (1.61)

T(g)v

=

([p.. . C))v.

The verification that T is an irreducible representation of G follows along the same lines as in the proof of Theorem (1.51). Thus we are lead to the following Theorem. (1.62) THEOREM. Let G be an abelian group with the direct sum decomposition G = (m,)@. (m,)

-

into cyclic subgroups (mi)of order m i , 1 5 i 5 r, where ail a i + l , 1 5 i r - 1. Then there is a one-to-one correspondence between the irreducible representations of G and the set of all r-tuples (it,. . . , [,), where Ci is an aith root of unity. If an irreducible representation T with representation space (v) corresponds to the r-tuple (Cl, . . . , C,) and g denotes j,m, + . . . + j , m,, then

T(g)v = ( C l j l . . . (‘3v. W e now turn to some specific instances of the representations theory of finite abelian groups.

82

2. The Representation Theory of Finite Groups

(1.63) EXAMPLE.We illustrate these general considerations by means of members of two of the three classes of isomorphic abelian groups of order eight. The Cayley table of the first of these two groups is as shown in Table ( 1.64). (1.64)

CAYLEY TABLEOF Cs 12345678 112345678 223456781 334567812 445678123 556781234 667812345 778123456 881234567

The group C , is a cyclic group of order eight which is generated by the element 2, for instance. From Theorem ( l . S l ) , it follows that the representations of C , are determined by the eighth roots of unity. Let o denote the complex number cos(ni4) + i sin(n/4). Then each of the numbers o,. . . , w 8 = 1, determines a distinct irreducible representation of C , . The complete collection of nonequivalent irreducible representations of C, is given by Table (1.65). (1.65)

IRREDUCIBLE REPRESENTATIONS OF Ce 1 2 3 4 5 6 7 8 T I 1 w w 2 w 3 w4 w 5 w 6 w7

T2

1 w 2 w4 w 6 w 8 w10 w I 2 w14

T3 1 w 3 w 6 w 9 w 1 2 w 1 5 w ' 8 w z 1 T~ I w4 w i z w i 6 w 2 0 w 2 4 w2a T5 I w 1 0 w 1 5 w Z O w Z 5 w30 w 3 5 T6 1 w 6 w 1 2 w 1 8 & 2 4 w 3 0 w 3 6 4,2,, T1 1 w 7 w14 w 2 1 w 2 8 w 3 5 w 4 2 w49 T a l l 1 1 1 1 1 1

The entries in the body of Table (1.65) constitute the matrices of the corresponding linear transformation with respect to any given basis of the one-dimensional representation space. (1.65') EXAMPLE.The next group G to be discussed is C, 0 C,. The members of C2 form the set (1, a} and those of C, the set (1, b, b2, b3}.The elements of C, @ C, are enumerated in the following order: 1 = (1, l), 2 = ( I , b), 3 = (1, bZ), 4 = (1, b3), 5 = ( a , I), 6 = ( a , b), 7 = ( a , b2), and 8 = ( a , b3).

83

I . Basic Concepts and Definitions

The Cayley table of G is shown in Table (1.66). The representations of C2 (1.66)

CAYLEY

TABLE OF

c z @ c4

12345678 112345678 223416785 334127856 441238567 556781234 667852341 778563412 885674123

are determined by the square roots of unity, 1 and -1, while those of C4 are determined by the fourth roots of unity, 1, i, - 1, and - i. If x and 6 are second and fourth roots of unity determining certain irreducible representations of C2 and C, ,then the pair {x,S} determine an irreducible representation T of C, 0 C, through the formula T[(u’,b ’ ) ] ~= xiS’v where {v} is a basis of a one-dimensional representation space V of T. These remarks lead to the family of eight irreducible representations of C, 0 C, which are listed in Table (1.67). The entries in the table are the matrices of the image Ti(gj)with respect to any basis of the representation space. (1.67)

IRREDUCIBLE REPRESENTATIONS OF Cz @ C4 1 2 3 4 5 6 7 8 1 i - 1 -i 1 i - 1 --i 1 i - 1 -i-1 -i 1 i 1-1 1-1 1-1 1-1 1-1 1 - 1 -1 1-1 1 T5 1 - i - 1 i 1 -i-1 i Ta 1 - i - 1 i-1 i 1 --i T ~ 1 1 1 1 1 1 1 T8 1 1 1 1 - 1 -1 - 1 -1

TI Tz T3 T4

1

These examples conclude the discussion of the representation theory of finite abelian groups. We turn now to the problems of the representation theory of nonabelian groups. It will appear that here the problems are considerably more complicated and require the introduction or a rather extensive mathematical apparatus.

84

2. The Representation Theory of Finite Groups 2. THE GROUP ALGEBRA KG OF A FINITE GROUP G

The results of Section 1 provide a powerful tool for the analysis of the representation space V of a linear representation T of a finite group. The principal purpose of this section is to develop a certain fundamental representation of an arbitrary finite group G. The immediate source of this representation is the group G itself. The usual representation obtained by means of G is called the regular representation and is denoted by the symbol 93 in this book. The representation space of the regular representation 93 is the group algebra KG of the finite group G. The group algebra KG is a vector space whose basis is identified with the elements of G and in which an operation of multiplication has been defined. Before entering into details, we summarize briefly the definition of the system known as an algebra in modern mathematics. Each algebra A is a vector space over a field K generally taken to be the field of complex numbers in this book. The symbol A is used to denote the set of vectors as well as the algebra itself. The connection between the algebra A and the field K is made explicit by saying that A is an algebra over K. In addition to the vector space operations of A, there is defined a binary operation called multiplication for the vectors. Thus there are three operations involved in the concept of an algebra: the operation of scalar multiplication, the operation of vector addition, and the operation of vector multiplication. In the following list of axioms, the symbols CI,P, and y denote elements of the field K of complex numbers, while the symbols x, y, and z are used for vectors of A. The symbol 1 denotes the multiplicative identity of K.

VECTORSUM (A Binary Operation on A) (1) x t (y 2 ) = (x Y)+ z. (2) There exists a unique 0 in A such that x 0 = 0 x = x. (3) Corresponding to x in A there is a unique y in A such that x y -tx = 0, where y is usually written -x. (4) x + z = z + x .

+

+

+

SCALARMULTIPLICATION (5) yx belongs to A. (6) 4 b x ) = ( a m . (7) a(x Y) = fix cry. (8) 1. P I X = ax Px. (9) Ix = x.

+ +

+ +

VECTORPRODUCT (A Binary Operation on A) (1) X(YZ = ( X Y N (2) 4XY) = ( 4 Y = X(UY>, (3) x(y 2 ) = xy -txz. (4) (Y z)x = yx zx.

+

+

+

+

+y =

2. The Group Algebra KG of a Finite Group G

85

The group algebra KG of a finite group G is an algebraic system that satisfies the above axioms and which is closely related to the group G. The initial step in its construction is to form a vector space associated with G. This may be done in the following way. Given any nonempty set S , the set F ( S ) of all complex valued functions with domain S and range K, the complex numbers, can be made into a vector space by means of the definitions: (i) Vector sum: Iff, g E F ( S ) , f + g is the element of F ( S ) whose value at s E S is given by [f+ gJ(s)= f ( s ) g(s). (ii) Scalar multiplication: I f f E F ( S ) and a E K, afis the element of F( S) whose value at s E S is given by [~fJ(s) = a(f(s)).

+

The argument that F(S) is a vector with these two operations is straightforward. In the case of a finite set S, each memberfof F ( S ) can be described by means of a table such as

where the first row of the table is an enumeration of the elements of S and the second is a tabulation of the images such that each image ci stands beneath its counter image s i . A second common mode of description is to write the element f of F ( S ) as a formal sum where the coefficients f ( s i ) of each element si is the value o f f at si. This notation can be abridged to (2.2) or

f=

n

1f(si)si

i= 1

These formal sums can be made into actual sums in F ( S ) by the identification of an element s of S with that function s* of F ( S ) whose value at s is one and whose value at s', different from s, is zero. With this notation, the above equations become (2.2')

or (2.3') We usually employ the notation of the unprimed equations, but think in terms of the primed equations. The vector space F(G), G a finite group, admits of a multiplication under which it becomes an algebra. The functions

86

2. The Representation Theory of Finite Groups

si* form a basis for the vector space F(G). A vector multiplication for F(G) can be defined in terms of these basis elements and extended to the other members of F(G) by linearity. Let si and sj be any two elements of G whose product s i s j is the element sk . We define

si*sj* = Sk*.

(2.4)

It follows that if h is a second element of F(G) with

h

=

c h(t)t,

isG

then the product f h should have the form

which becomes, after substituting q for st and rearranging,

This result justifies the definition: (iii) Vector product: Iff, h E F(G),fh i s the element of F(G) whose value at s E G is given by [fh](s) = ~,,Gf(st-1)17(t). The proof that F(G) i s an algebra over the field K of complex numbers with these definitions of vector sum, scalar multiplication, and vector product is a tedious argument which we advise the reader to skip. The algebra F(G) is called the group ulgebra of the group G and denoted in the sequel by KG. There is associated with each element b of an algebra A an element b, of the K-homomorphisms Hom,(A, A) of A into itself. The definition of bL is straightforward. If x E A, then bL(x) is defined by bL(x) = bx.

The fact that b, is linear follows from the observations that and

bL(x + y)

= b(x

b,(r*x)

+ y) = bx + by = bL(X) + bL(y),

= b(ctx) = a(bx) = a(bL(x)),

a

E

X,

y

E

A,

K, x E A.

The re~gulurrepresentation 3' 3 of the group G is the homomorphism of G into GL(KG) obtained by letting each image %(g) be the left-multiplication 9 , . a linear transformation on the vector space (algebra) KG. The symbol gL denotes the left-translation or left-multiplication defined by the element g* of KG. Recall our custom of denoting the element g* by the element g of G to which it corresponds and, consequently, of denoting its left-multiplication by g L .

2.

87

The Group Algebra KG of a Finite Group G

We wish to verify that the mapping % is a representation of the finite group G. Let g , g’ be elements of G and x be any element of KG, then W g g ’ ) x = (SS’)L(X) = (99% = g(g’x) = SL(SL’(X)) =W)(%’)X)

so that (2.7)

= Pw)%(g’)lx,

W g d ) = f#(g)Wg‘).

Furthermore, we see that f l ( l ) X = lL(X) = l(x) =x = lKG(X),

which implies that %(1) = l K G ,

(2.8)

Equations (2.7) and (2.8) show that the mapping % is a homomorphism of the finite group G into GL(KG) so that the term regular representation is justified, that is, % is a linear representation of G over the representation space KG. Before developing the structure theory of the group algebra KG of a finite group G, we consider a simple example in some detail. There are two classes of abstract groups of order six, one of which is abelian and the other is not. The permutation group S3 is a realization of the noncommutative class whose elements may be defined as follows: R1

a b c = ( a b c)’

R2 =

a b c ( c a b)’

a b c R 3 = ( b c a)’

a R4=(a

b c c b)’

a b c R 5 = ( b a c)’

a b c R 6 = ( c b a)‘

The multiplication table, group elements being designated by their subscripts, for the group S3 is shown in Table (2.10).

(2.10)

CAYLEY TABLE OF S3 123456 1123456 2231645 3312564 4456123 5564312 6645231

88

2. The Representation Theory of Finite Groups

The entry k at the intersection of the ith row and j t h column is the number of the product R, = Ri R, of the element R i by the element R, . The elements in KG which correspond to the group elements form the set &, R 2 , R 3 , R 4 , R 5 , R6)? where R2

R3

Rt

R2

R3

R4

R5

R6

Rt

R2

R3

R4

R5

R6

R2

R3

R4

R5

R6

Rt

R2

R3

R4

R5

R6

Rt

R2

R3

R4

R5

R6

1

0

0

(

R1

0 0

0

1

0

(2.1 1)

R, 0

R4 R5 0 0

Rl

0

0

0

0

1

0

0

0

0

0

0

0

0

1

0

0

0

0

0

=R2, =R3,

0

1

0

= Rt,

0

0

1

The notation for these elements can be conveniently shortened to (1 0 0 0 0 0) = R,, (2.12)

(0 1 0 0 0 1 0 0 (0 0 0 1 0 (0 0 0 0 1 (0 0 0 0 0

(0 0

0)=R2, 0)=R3, 0)=R4, 0)=R5, 1) = R 6 ,

where the first row in each tab-" is understood to be Rl

R2

R3

R4

RS

R6

and is suppressed. The form of the expressions in (2.12) show that the vector space KS, is isomorphic to K 6 . In particular, {Ri},1 5 i I 6, is a basis for KS3 called the standard or natural basis. It follows from the definition of vector product, that

[RiRj](s)=

rsG

Ri(st-l)R,(t).

89

2. The Group Algebra KG ofa Finite Group G

This value is different from zero for a given S if and only if at least one of the summands on the right-hand side is different from zero. The summand R;(st-1)R/t) differs from zero if and only if and

st-1=Sj

t=Sj'

Thus [RjR¥s) is different from zero if and only if

Since [R j Rj](sj Sj) (2.13)

=

1, these observations imply that where

Ri R, =R k,

SjSj =Sk'

Thus we see that the rules of (2.4) are satisfied and that the multiplication in the group algebra KS 3 is completely determined by the group multiplication Table (2.10). In particular, the effect of the left-multiplication R iL is determined from Eq. (2.13). Each left-multiplication R iL , being a linear transformation, has a matrix M(i) with respect to the standard basis. The elements of M(i) are denoted by the symbols M(i)jk' Thus we have 6

RiL(R j) =

I 1 M(i)kj R k·

k=

The explicit forms of the matrices are 1 0 0 0 0 o 1 000 o 00100 o M(I) = 00010 o o 0 0 0 1 o 00000 1

o

o

'

1 000 o 0 100 o 1 0 0 0 0 o M(3) = o 0 000 1 ' 00010 o o 0 0 0 1 o

o o

00001 o 0 000 00010 M(5) = 00100 1 0 0 0 0 o 1 000

o I

o o' o o

0 I 0 0 10000 o 1 000 M(2) = 00001 00000 00010 00010 o 0 0 0 1 o 0 0 0 0 M(4) = I 0 0 0 0 o 1 000 00100

o 0 0 0 0 000 I 0 o 0 0 0 I M(6) = o 1 000 o 0 1 0 0 1 0 0 0 0

o o o o' 1

o o o I

o' o o 1

o o o o o

90

2. The Representation Theory of Finite Groups

We return to the general considerations. The theory of algebras employs certain basic concepts which can be applied to a group algebra KG as a special case. (2.14) DEFINITION. A right ideal J of an algebra A is a subspace of A such that if x E J, y E A then xy is an element of J. A left ideal J of A is a subspace of A such that if x E J, y E A, then yx is an element of J. A two-sided ideal J of A is a subspace which is both a right and a left ideal.

In every algebra A, the subspace A and the zero space (0) are ideals referred to as fricial ideals. Ideals of A different from either of these are called nontricial ideals. A ready example of these concepts is had in the case of the algebra K, of all 2 x 2 matrices over the complex field. The argument that K2 is an algebra is essentially a review of the basic operations on matrices. The set of all matrices of the form

: :1 :I1 1 at 1

is a left ideal in K 2 , while the set of those of the form " ; ;

is a right ideal. It is an interesting fact that K2 contains no nontrivial twosided ideal. For let J be any two-sided ideal of K2 which contains a nonzero element

The products

/I2 2 1 . )I ::: ::: 1 / ; ;/ 1 ::: :::/ 1 0 I/ I( ::: :: /I / ::; 2: /I =

and

I ! )

=

belong to J. The first is the original matrix with its columns interchanged and the second with its rows interchanged. It follows from a sequence of such operations that J contains a matrix B with b,, different from zero whenever J contains a nonzero element of K 2 . On the other hand, it follows from

91

2. The Group Algebra KG of a Finite Group G

and the fact that J is a subspace, that J must contain the matrix

Continuing in this fashion, one finds that J must contain a basis of K2 consisting of the four matrices

Consequently, J coincides with the algebra K , .

(2.15) DEFINITION. A minimal ideal is an ideal which contains no nontrivial ideal of the same nature. A minimal left ideal contains no nontrivial left ideal, a minimal right ideal contains no nontrivial right ideal, and a minimal twosided ideal contains no nontrivial two-sided ideal. (2.16) LEMMA. A subspace L of the group algebra KG is a left ideal of KG if and only if L is an invariant or reducing subspace of the regular representation Y?. Proof. Let L be a reducing subspace of the regular representation 92, t be any element of G, and u be an element of L. Then Consequently, if h

=

ct

tu

= t,u = Y?(t)u E L.

h(t)t any element of KG, then

and L is a left ideal in KG. Conversely, if L is a left ideal of KG, g u E L, then %(g)u = gLu = gu E L

E

G, and

and L is a reducing subspace of Y?. We note that an irreducible subspace L of the regular representation % is a subspace of KG which contains no nontrivial reducing subspace. A minimal left ideal of KG is a left ideal containing no nontrivial left ideals. It follows as a corollary.

(2.17) COROLLARY. A subspace L of the group algebra KG is an irreducible subspace of the regular representation Y? if and only if L is a minimal left ideal of KG. We turn to the introduction of an important new idea. Since any left ideal L of the group algebra KG is also an invariant subspace of '93, the regular representation defines a representation sL whose representation space is L.

92

2. The Representation Theory of Finite Groups

Two left ideals L and L’ are said to be equivalent if their corresponding representations 91L and 91L, are equivalent. We denote this relation in symbols by L z L‘. (2. IS) THEOREM. Every left ideal L of the group algebra KG of a finite group G has a complementary left ideal L’ such that KG is the direct sum L @L’. Proqf: This is a special case of Theorem (1.19). (2.19) THEOREM. The group algebra KG of a finite group G is the direct sum of minimal left ideals:

KG = L , @ . . . @ I + .

(2.20) Furthermore, if

KG =L,‘ @... @L,‘

(2.21)

is another decomposition of KG as the direct sum of minimal left ideals, then k equals t and. possibly after rearrangement, the corresponding minimal left ideals are equivalent, that is,

Li M L j z ’ ,

(2.22)

1l ilk.

Proof. This is a particular instance of Theorem (1.45) for the case of the group algebra KG, where the term minimal left ideal is used rather than irreducible subspace. Observe that two minimal left ideals L and L’ of an algebra A either coincide or have only the zero vector in common. The sum L @ L’ is direct i n the second case. Furthermore. if L and L’ are distinct minimal left ideals of the group algebra KG, there exists a direct sum decomposition

KG = L @ L ’ @ * . . @ L k

(2.23)

into the sum of minimal left ideals. Our results and definitions have been concerned with representations of a finite group G up to the present. We enlarge our concepts to include the representations of an arbitrary algebra A. A linear representation of an algebra A is a mapping T* of A into the algebra Hom,(V, V) of all linear transformations of a vector space V over the field K of complex numbers. The mapping T* preserves the algebraic operations in the following sense. Let cx denote a complex number while x and y denote elements of the algebra A. Then

+

(i) T*(x y) = T*(x) (ii) T*(ax) = cxT*(x);

+ T*(y);

(iii) T*(xy) = T*(x)T*(y).

93

2. The Group AIgebra KG of a Finite Group G

Although the preceding definition applies to the concept of a representation for any algebra, we use it primarily in the case of the group algebra KC of a finite group. One notes that a representation T* of an algebra A is a linear transformation of the vector space A into a vector space Hom,(V, V). This contrasts with the case of a group representation T which is homomorphism of one group into another. The analysis of the representations of certain special types of algebras is more straightforward than the corresponding analysis of the representations of an arbitrary finite group. Consequently, some of our representation problems are simplified by noting that every linear representation T of a finite group G defines a corresponding representation T* of the group algebra KG. Let T be a linear representation of the finite group G on the finite dimensional vector space V over the field K of complex numbers. The representation T* is defined in the following manner. The set

G

= {gi, . . . , g,>

is a basis of the vector space KG. The set Im T = {T(gi),. . . , T(g,)) belongs to the vector space Horn#, V). By a well-known theorem of vector spaces, there exists a unique linear transformation T* with domain KG and range Hom,(V, V) such that T*(gJ = Tfgi),

(2.24)

9, E G .

In Eq. (2.24), the argument of T* on the left should be labeled g , * ; however, we eliminate the asterisk from such elements of KG by agreement. The argument of T on the right-hand side of (2.24) is an element of G. Since T* is known t o be linear, it suffices to show that

T*(xJJ) = T*(x)T*(y),

(2.25) Let f =

1,

f(g)g and Iz =

cr

f / f

from which it follows that

=

X,

y

E

KG

I7(t)t be any two elements of KG. Then

c c fw7(f)sf>

gsc

rsc

94

2. The Representation Theory of Finite Groups

We see that T* is a linear transformation of KG into Hom,(V, V) which satisfies Eq. (2.25) listed above. Therefore, T* is a linear representation of the group algebra KG on the representation space V. Furthermore, if T' any representation of KG such that (2.27)

T'(d =

m,

9

E

G,

then T' coincides with T*. On the other hand, let T* be any linear representation of the group algebra KG on the representation space V, with T * ( l ) the identity element 1, of Hom,(V. V). Then the mapping T of G into Hom,(V, V) defined by (2.28)

T(g) = T*(g),

9

E

G,

is a representation of the group G. Let T be a representation of the finite group G and T* a representation of the group algebra KG, related to each other according to one of the two schemes outlined above. Then T and T* have a common representation space V. Furthermore. the concepts of an invariant or reducing subspace U coincide for the two representations; that is, a subspace U is a reducing subspace of T if and only if it is a reducing subspace of T*. It follows immediately from this fact that the concepts of irreducible, decomposable, and completely reducible coincide for the two representations T and T*. The representation space of the regular representation % of the finite group G is none other than the group algebra KG. It follows that the corresponding representation %* of the group algebra KG is a representation of the group algebra KG with representation space also KG. We will show that the representation %* makes correspond to every f E KG the left-multiplication fL of Hom,(KG. KG). To see this, let f be any element of KG and v be some element of KG (considered as the representation space of %*). Since f can be expressed in the form

f=

c f(d%

SEG

we see that

In summary, (2.30)

!R*(f) =f L ,

f E KG.

95

2. The Group Algebra KG of a Finite Group G

The content of the preceding remarks is that the representation %* of the group algebra KG can be viewed consistently either as an extension of the regular representation % of the group G or as the regular representation of the algebra KG which makes correspond to each f of KG the left-translation f L. We now turn to an investigation of the representations of the group algebra KG of a finite group G over the field K of complex numbers.

(2.31) THEOREM. Every irreducible representation T of the group algebra KG occurs in the regular representation %* of KG. Proof. Denote by !RL* the representation of KG defined by %* on the minimal left ideal L of KG. Let T be any irreducible representation of the group algebra KG with representation space V. Define a linear transformation A of a minimal left ideal L of KG into the space V in the following manner. Select any nonzero vector v of V and denote by A the linear transformation of L into V defined for x E L by A X = T(x)v. Let f be any element of KG and note that whenever v E L [A%L*(f>lX= A[%L*(fBl = 4 f x I

= T(f"x1

=

[T(f)AIx.

It follows that

(2.32)

T ( f ) A = A91L*(f),

f

E

KG.

Since both 91L* and T are irreducible, Schur's lemma implies that either A is the null transformation or else A is an isomorphism so that sL* is equivalent to T. Suppose that for some irreducible representation T, A is the null transformation for each choice of minimal left ideal L of KG. Since KG is the direct sum KG=Ll@*..@Lk of minimal left ideals, it follows that T ( f ) v must be zero for every f E KG, which is a contradiction. Hence the group algebra KG must contain at least one minimal left ideal L such that %* is equivalent to T. In the sequel, the regular representation %* of the group algebra KG is denoted by the same symbol % as the regular representation of the finite group G unless there is serious danger of confusion about the meaning of the symbols.

(2.33) THEOREM. Let T be any linear representation of the group algebra KG over a finite-dimensional representation space V and let T have the decomposition T = T1 0 ' . @ Tk

96

2. The Representation Theory of Finite Groups

into irreducible components. Let x belong to a minimal left ideal L of KG such that 9IL is not equivalent to any of the summands Ti, 15iI k. Then it follows that T(x) must be the zero linear transformation. Proof. It was seen in the proof of Theorem (2.31) that if the irreducible representation T iis not equivalent to the representation !RL, then Ti(x) is zero whenever x E L. Since T(x) = T,(x) . . . Tk(x), it follows that T vanishes for x E L whenever !RLis not equivalent to any one ofthe T i ,1 5 i 5 k.

+ +

3. THE STRUCTURE OF THE GROUP ALGEBRA KG

The fundamental decomposition of a group algebra KG into the direct sum of minimal left ideals was obtained in Section 2 . I n Section 3, it is shown that the group algebra KG of a finite group G over the field K of complex numbers is the direct sum of minimal two-sided ideals. Every complex group algebra KG i s a sernisirnple algebra, a concept to be introduced in this section. The development of these results requires the introduction of a number of new algebraic ideas. A left ideal L of an algebra A is the principal left ideal generated by the element b of A whenever L coincides with the left ideal Ab. The element b is a generator of the ideal L. A nonzero element e of A such that ez equals e is an idernpotent. If the left ideal L has the idempotent e for a generator, then e is an idernpotent generator of L. Let 6 be any nonempty subset of the algebra A . The set L of all a E A such that as is zero when s E 6 is a left ideal called the left annihilator of 6 . The set R of all a E A such that sa is zero whenever s E 6 is a right ideal called the right annihilator of 6 . (3.1) THEOREM. Let the group algebra KG of the finite group G over the field K of complex numbers be the direct sum L @ L’ of the nontrivial left ideals L and L‘. Then there exists idempotents e and e‘ which are the generating idempotents of L and L’, respectively. Furthermore,

ee‘ = e‘e = 0.

(3.2)

Proof. Let 1 denote the multiplicative identity in KG. Then there exists the decomposition

1 = e + e‘,

If x is any element of L, then

x

= x l = xe

+ xe’,

e E L , e‘ EL’. xe E L, xe’ E L’.

From the uniqueness of the direct sum decomposition of x, we see that

x =xe,

0 =xe’

97

3. The Structure of the Group Algebra KG

In particular, it follows that e = e 2.

O=ee'.

In a similar fashion, one has for x' any element of L', x'

= x'e',

0

= x'e,

and

e'

= (e')',

0

= e'e.

Also one notes that every x E L can be written x = xe so that L c (KG)e. Since (KG)e is surely contained in L, it follows that L coincides with (KG)e. Thus e is an idempotent generator of L. In the same way, e' is an idempotent generator of L'. (3.3) COROLLARY. If L is a nontrivial left ideal of K C , then L has an idempotent generator e such that the right-multiplication eR, see (3.9), is a linear transformation of KG onto L. Moreover, xe and x coincide for every x in L. Proof: Theorem (2.18) asserts that KG is the direct sum L @L' where L' is an ideal complementary to L. Denote by e and e' the idempotents of Theorem (3.1). The element e is the required idempotent generator of L and eR is the required linear transformation. Let the group algebra KG be the direct sum (3.4) THEOREM. (3.5)

KG = L l @ . . . @ L k

of nontrivial left ideals {L,}, 1 5 i 5 k. Then there exists a family of idempotents { e , } ,1 I iI k , such that each e , I S a generating idempotent of the corresponding left ideal Li . Furthermore, (3.6)

e,e,

= 0,

i #.j.

Proof: Decompose the identity 1 of KG as the sum

(3.7)

1 = el

+ . . . + ek

and proceed as in Theorem (3.1). Two idempotents e' and e" are called orthogonal if e'e''

= e"e' = 0.

An idempotent e is called primitive if there exists no decomposition of e into the sum of orthogonal idempotents. (3.8) THEOREM. If e is a primitive idempotent of the group algebra KG, then the left ideal L generated by e is minimal. Conversely, if L is a minimal left ideal of KG, then any idempotent generator e of L is primitive.

98

2. The Representation Theory of Finite Groups

Proof Let e denote an idempotent generator of the left ideal L. Suppose that L’ is a nontrivial left ideal properly contained in L, then KG and L have the respective direct sum decompositions,

KG

= L’ @ L”

and

L = L’ @ L*,

where L* denotes the nontrivial left ideal L n L”. The idempotent e has the decomposition e = e‘

+ e*

with e’ and eh orthogonal idempotents generating L’ and L*, respectively. Thus e is not primitive. Let e be a nonprimitive idempotent with the decomposition e’ e“ into the sum of orthogonal idempotents. The left ideal generated by e has the decomposition

+

Le = L(e’ + e”) = Le’ + Le”. The ideals Le’ and Le” are contained in Le since

e’e = e’(e’ + e”) = (e’)’

= e‘ E

Le

and

e“e = e”(e’ + e”) = (e”I2= e“ E Le. However, Le’ does not coincide with Le since e“ belongs to Le but not to Le’. Thus L is not a minimal left ideal. (3.9) OBSERVATION. Every element b of an algebra A over the field K of complex numbers determines a right-multiplication (translation) b, belonging to Hom,(A, A). If x any element of A, then bR(x) is defined by

(3.10)

bR(x) = xb.

The argument that the right-multiplication b, is a h e a r transformation parallels the proof that the left-multiplication bL is a linear transformation. However, the homomorphism bR has an additional property which is not always valid for a general linear transformation. This property is as follows: If a E A. considered as an algebra, and x E A, considered as a vector space, then

b,(ax) = (ax)b

= a(xb) = a(b,(x));

that is, (3.11)

b,(ax)

= a(b,X).

99

3. The Structure of the Group Algebra KG

Let V and W be A-modules and let h be a group homomorphism of V into W. Suppose that h(ax) = a/z(x),

(3.12)

a

E

A, x

E

V.

Then h is called an A-homomorphism of V into W. The collection of all Ahomomorphisms of V into W is denoted by the symbol Hom,(V, W). Equation (3.11) asserts that bR is an element of Hom,(V, V) as well as an element of Horn#', V). We wish to emphasize the fact that the two left ideals L and L' of the group algebra KG are said to be equivalent if and only if the two representations '91L and 'illLf are equivalent. (3.13) THEOREM. If the left ideals L and L' are equivalent, then every equivalence mapping A from L into L' is given by a right-multiplication, that is, there exists b E KG such that A(X)

= bR(X)= xb,

X E L.

Proof. The statement that L and L' are equivalent implies that there exists an isomorphism A of L onto L' such that A%L(f)

= '9IL'(f)A,

fE

Let e be a generating idempotent of L. Then x

KG.

EL

implies that

xe = x .

It follows that A(x)

= A(xe) = A('91L(x)e)=

[A'illL(x)]e = ['91L.(x)A]e= %,.(x)(Ae) = %,.(x)b = xb = bR(X),

where b is the image Ae of e under A and b, is the right-multiplication defined by b. (3.14) REMARK. The element b determined above has the property that b equals eb. Furthermore, since b is an element of L', b is equal to be'. Thus the equation b = eb = ebe'

is valid. It is significant that any nonzero element y for which y

= eye'

defines a nonnull right-translation yR of L into L'.

100

2. The Representation Theory of Finite Groups

We turn to the consideration of certain general facts which can be conveniently expressed in terms of the concept of KG-homomorphism. (3.15) LEMMA.Let h be a KG-homomorphism of the minimal left ideal L into the minimal left ideal L’. Then h is either an isomorphism or the null homomorphism. Proof: This is Schur’s lemma (1.13) with a slight variation. Since h E Hom,,(L, L’), / I is also an element of Hom,(L, L’). Furthermore,

/?(ax) = a/7(x),

a

E

x

KG,

E

L,

so that (3.16)

PL(a)x) = sL,(a)(~W)

or (3.17)

[/?%L(a)l(x)= [ f l , W ~ l ( X ) .

Equation (3.17) implies that

’iNL3(a)/7= hSL(a),

a

E

KG.

Since both L and L’ are irreducible spaces for 91, it follows from Schur’s lemma that 17 must be either an isomorphism or the null homomorphism. The second form of Schur’s lemma has the following interesting consequence.

(3.18) LEMMA.Any KG-endomorphism h of a minimal left ideal L into itself is either an automorphism of L or the null endomorphism. ProoJ The proof consists in redefining the terms so as t o apply Lemma ( 1.1 5 ) . (3.19) THEOREM. Let L and L’ be minimal left ideals with generatingidempotents e and e‘ respectively. Then any element exe‘, different from zero, defines an equivalence mapping of L onto L’. Proof. The right-multiplication bR defined by a nonzero element b = exe‘ is not the nullhomomorphism since bR(e) = b. By Lemma (3.15), bR is a KG-isomorphism of L onto L’. It follows that if x E L then (3.20)

[bRflL(f)I~= bR[% ( f B 1 = bR(f~) = (fx)b =f(xb) = f ( b R ~ ) = sL’(f)(bRX) = [%L’(f)bRlxConsequently, bR%(f)

= fl~,(f)bR,

f e KG.

Since b, is not the null mapping, the representations 91, and SLr are equivalent, that is. the minimal left ideals L and L’ are equivalent.

3. The Structure of the Group Algebra KG

101

(3.21) REMARK.It is only asserted in (3.14) and (3.19) that the equivalence can be effected by an element of the form exe‘, not that it necessarily is effected by such. Nevertheless, if b is any element determining such a n equivalence by right-multiplication, then eb is a nonzero element of L’ so that an element of the prescribed form, namely, eb = ebe‘, also defines the same right-multiplication of L onto L‘. Thus we have the following theorem. (3.22) THEOREM. Two minimal left ideals (KG)e and (KG)e’ are equivalent if and only if there exists an element exe‘ different from zero. Every equivalence mapping from (KG)e onto (KG)e’ is given by a right-multiplication defined by such an element. We now present a fundamental theorem on primitive idempotents. (3.23) THEOREM. If e is a primitive idempotent of KG, then exe is a multiple of e for every x E KG. Conversely, if exe is a multiple of the idempotent e for every x in KG, then e is a primitive idempotent. Proof: Every primitive idempotent e is the generator of a minimal left ideal L of the form (KG)e. If exe is null, then it is of the form Oe. Any nonzero element b of the form exe determines a right-multiplication b, of L into L such that

f E KG.

‘ % , ( f ) b= ~ bR’%(f)-

(3.24)

Since !RL is irreducible, it follows from Schur’s lemma that b, must be a multiple of the identity transformation on L. Consequently, In particular,

x(exe) = ~ x , x E L , e(exe)

= exe = Xe,

xEK E

K.

Conversely, let e be an idempotent of KG such that Suppose that with

(e’)2 = e‘,

Then, it follows that Consequently,

exe = Xe, e = e‘

x

E KG.

+ e“,

(e”)’ = e”,

and

e’en= e”e’ = 0.

e’ = ee’e = ze. Xe‘ = X2e = (xe)’

= (e’)’ = e’.

Therefore, the number x must be either 1 or 0. It follows that e‘ either equals e o r zero. Thus e is primitive.

102

2. The Representation Theory of Finite Groups

(3.25) THEOREM. Let the group algebra KG of a finite group G be the direct sum J @ J’ of two-sided ideals, J and J’. Then J and J ’ annihilate each other; that is, x E J, x’ E J’ imply that xx’ = x’x = 0. Furthermore, J and J’ have unique idempotent generators e and e’ respectively, each of which commutes with all the elements of KG.

Proof. The element 1 of KG has the decomposition e + e’, where e belongs to J and e’ belongs to J’. It follows from Theorem (3.1) that xe,

0 = xe’,

x E J,

xf = x’e’,

0 = x’e,

x‘ E J’.

x =ex,

0 = e‘x,

x

x‘ = e‘x‘,

0 = ex’,

x’ E J‘.

x

=

and In the same way, E

J,

and From these relations,

ee’ = e‘e = 0 and

xx’ = xex’ = 0

= x’e’x = x‘x,

whenever x E J and x’ E J’. We note, in particular, that e is a two-sided identity for J and that e’ is a two-sided identity for J‘. Finally, any y E KG can be written x x‘, where x E J and x’ E J’. We have

+

ey

= e(x

+ x’) = ex = xe = (x + x’)e = ye,

so that e is an idempotent in the center of KG. Similarly, e’ is an idempotent in the center. If e and e* are idempotent generators of J, then ee* is e and e*e is e*. Consequently.

e

= ee” = e*e = e*.

Thus e is unique (3.26) THEOREM. Let J be a two-sided ideal and L be a minimal left ideal of the group algebra KG. Then either L n J = L

or

LnJ=(O).

Proof. Since the minimal left ideal L contains the left ideal L n J, it follows that L n J must either be (0) or L itself.

103

3. The Structure of the Group Algebra KG

Thus, if KG is the direct sum J @ J’ of two-sided ideals J and J’, then each minimal left ideal L of KG is either contained in J or in J’. (3.27) THEOREM. If the two-sided ideal J of the group algebra KG contains the minimal left ideal L, then J contains every minimal left ideal L’ equivalent to L. Proof: We have shown that the minimal left ideals L and L’ of KG are equivalent only if there exists a right-multiplication b, of L onto L . Therefore L is contained in the two-sided ideal J iff

L’ = Lb,

b E KG,

is also contained in J. Let the group algebra KG of a finite group G be the direct sum (3.28)

KG=L,@...@L,

of minimal left ideals. Denote by J, the direct sum of all summands of (3.28) which are equivalent to L,, , where Li, is L, ; denote by J2 the direct sum of all summands of (3.28) equivalent to L,, , where Lizis the first summand of (3.28) not occurring in J, ; and, in general, denote by J, the direct sum of all summands which are equivalent to Lis where Li3 is the first summand of (3.28) not occurring in the sum

J, @ - * - @ J , - , . This construction leads to a decomposition of KG as the direct sum KG

= J, @ . - .@ J,

of left ideals J i , 1 I iI m . We wish t o show that each left ideal J i of this decomposition is also a right ideal. Let J, contain the nonzero element x belonging to some minimal left ideal L,, with idempotent generator e,, and let y denote any element of KG. Then (3.28’)

(xe,)y

= xy = x1

+ ... + x k ,

where each x, of (3.28’) belongs t o the summand L, of (3.28). If x, differs from zero, then x(e, ye,) = (xy)e,

= (xl

+

* * *

+ xk)er= x, .

Consequently, e, ye, is different from zero which implies that the minimal left ideal L, is equivalent to the minimal left ideal L, belonging to J, . However, Theorem (3.27) asserts that L, is contained in Ji under these circumstances. Therefore, xy belongs to Ji. Since each element of J, is a sum of elements of the type just considered, it follows that J i is a right and, consequently, a

104

2. The Representation Theory of Finite Groups

two-sided ideal. Any nonzero two-sided ideal J contained in J i must contain at least one minimal left ideal L of J , . By Theorem (3.27), J must contain every minimal left ideal L' equivalent to L and, consequently, must contain J , . Therefore, J, is a minimal two-sided ideal of KG. We are lead to the following theorem. (3.29) THEOREM. The group algebra KG of a finite group G can be decomposed into the direct sum KG = J , @ - * * @ J , of minimal, two-sided ideals in essentially one way. Proof: The argument that such a decomposition exists has been given. Suppose that

KG=J,@.**@J,

(3.30) and

KG

(3.31)

= J,' @ . . . @ J,'

are two direct sum decompositions of the group algebra KG. We wish to show not only that the number of summands must be the same in each of the sums (3.30) and (3.31), but also that the same summands must appear, although perhaps in different order. First, observe that if J and J' are minimal, two-sided ideals, then J n J' is a two-sided ideal contained in each. Therefore, J n J' must either be (0) or else must coincide with J and J'. The multiplicative identity of KG has the expansion l=e,+...+e,,

(3.32)

where e, is an idempotent generator of J i , 1 5 i I k. It is left as a problem to show that each e, is the only idempotent generator of J, and that each e, belongs to the center of KG, 1 i i I k . There is also a decomposition

1 = el' +

. + et',

with each summand e,' the unique, central idempotent generator of J i , 1 5 s 5 t. Since each such e,' can be expressed in the form, e,'

= e,'

1 = es'el + . . . + es'ek,

it follows that, for some e k ,e i e k is a nonzero element of J,' n J,. Conse-

quently, J,' and J, coincide. Therefore, every summand of (3.31) is a summand of (3.30). A similar discussion shows that every summand of (3.30) must be a summand of (3.31), which completes the argu,nent.

We pause to introduce several new concepts. Let A be a finite-dimensional algebra over the field K of complex numbers. An ideal J of the algebra A is

3. The Structure of the Group Algebra KG

105

said to be nilpotent if there exists a positive integer n such that any product of more than n factors from J is always zero. The sum of all the nilpotent left ideals of the algebra A is a left ideal N called the radical of A. It can be shown that N is a two-sided nilpotent ideal of A and that any nilpotent ideal of A is contained in the radical N. (3.33) THEOREM. The radical N of the group algebra KG of a finite group G over the field K of complex numbers consists of zero alone. Proof. The group algebra KG is the direct sum, KG = N O ” , of the radical N and a complementary left ideal N’. The multiplicative identity 1 of the group algebra KG can be expressed as the sum, l=e+e’,

e E N , e’EN’,

where e2

= e,

(el)’

= e’,

and

ee‘ = e’e = 0.

Consequently, the equality en = e

holds for every positive integer n. It follows, since N is nilpotent, that e is zero and that 1 belongs to N’. Every element x of KG is of the form xl, an element of N’. Consequently, the radical N of KG is the zero ideal, as was to be shown. We turn t o another important definition. (3.34) DEFINITION. Let A be an algebra with an identity over the field K of complex numbers. The algebra A is said to be semisimple if and only if (i) the radical N of A is the zero ideal, and (ii) the left ideals of A satisfy the descending chain condition. A semisimple algebra A which contains no nontrivial, two-sided ideals is said t o be simple. In Definition (2.14), each left ideal L of an algebra A over the field K is defined to be a K-subspace of A which is closed under left-multiplication by any element of A. When the algebra A contains a multiplicative identity 1, a second, equally satisfactory definition, is merely that each left ideal L is only a subgroup, closed under left multiplication. In this instance, the set K’ of all elements of A of the form ctl, ct E K, is a subalgebra isomorphic to K. This subalgebra K ’ can be identified with K so that K becomes a subalgebra of A. Under these circumstances, the left ideals become left Ksubspaces which means that the left ideals of the second definition are left ideals according to the first. Since a group algebra KG has dimension [G : 11,

106

2. The Representation Theory of Finite Groups

it follows that both the ascending and descending chain conditions hold for

subspaces and hence for left ideals (using either definition of left ideal). According to Theorem (3.33), the group algebra KG of a finite group G over the field K of complex numbers has zero radical. Thus, such a group algebra KG is a semisimple. In passing, it is useful to remark that this result does not necessarily hold for group algebras over other fields. In the decomposition (3.30), the minimal two-sided ideals; J,,1 I i 5 k ; are simple algebras which are called the simpIe components of KG. The proof of the simplicity of these ideals is left as a problem. 4. THE SIMPLE COMPONENTS OF THE GROUP ALGEBRA KG

The decomposition of the group algebra KG into the direct sum of its simple components has been obtained in Section 3. In Section 4, it is demonstrated that each simple component i s isomorphic to an algebra K, of all n x n complex matrices for some positive integer n. This analysis leads to the result that the group algebra KG of a finite group G over the field K of complex numbers is isomorphic to an algebra A of complex matrices, each of which appears in the same quasi-diagonal form. The number of diagonal blocks in the pattern is equal to the number of minimal, two-sided ideals appearing in the decomposition (3.30). Denote any one of the summands of Eq. (3.30) by the symbol J. Let the sums

(4.1)

J=L,@”*@L,

and

e=e,+.-.+e,

be the decompositions of J and its generating idempotent e according to some set of equivalent minimal left ideals of J. Remember that (4.2)

L, = Je, = (KG)ei,

1 <_ i I k.

Since the minimal left ideals of J are mutually equivalent, there exists a subset of J, B = {bij},

1 _< i, j I k,

such that each b,, , a nonzero element of the form eixej, determines a right multiplication of L, on L,. Furthermore, bii can be taken to be the given e , , 1 5 i 5 k. The set

V

= {eixej : x E

KG}

is a nonzero subspace of L,. The subspace V i s mapped into the onedimensional subspace W

= {e,xei: x E

KG}

107

4. The Simple Components of the Group Algebra KC

of Li by the right multiplication b,, b,Z = zbji,

Z E

KG.

Since b, is an isomorphism of Lj, it follows that V is a one-dimensional subspace of Lj ; that is,

V

(4.3)

= {V = ubij :

E K}.

Let x denote any element in Lj . Then one has the equalities x

(4.4)

= xej = exej = e,xej

+ . + ekxej. *.

This equation can be written in the form

(4.5)

X = 5 1j

b, j

+ + *

’’

lkjbkj,

where the set {tlj , . . . , t k jis } contained in K. Therefore, the set Bj of elements, 1I i I k,

{bij},

constitutes a set of generators of Lj. Suppose that 0

(4.6)

=t,jblj

Equation (4.6) implies that 0

+

’ * *

+

tkjbkj.

= erO = &jb,j,

according t o which, 0 =trj.

(4.7)

Consequently, the set Bj is a linearly independent set of generators of L,, that is, Bj is a basis of Lj. Every element y belonging to J can be written (4.8)

y

= eye = ey(e,

+

. + ek)

=elye, + . . . + e , y e , =?l,bii

+-**+

+“’+?kibkl

elyek+...+e,yek

+ “ ’ + ~ l k b l ~ + “ ‘ + ~ k ~ b k ~ .

This shows that the set B

= B, v

*..

v Bk

is a generating set of J. It is easy to prove that B is linearly independent over K so that it is a K-basis of J. In particular, one notes that the K-dimension of J is k 2 where k is the number of minimal Ieji ideals occurring in any direct sum, decomposition of J into minimal Ieft ideals. It proves possible t o select a set, M

= {eij},

1I i, j

s k,

108

2. The Representation Theory of Finite Groups

such that M is a basis of the minimal ideal J with the multiplicative properties (4.9)

eijers = ajreis,

for all permissible values of i, j , Y, and s. As a matter of fact, there exists a set {vij}of multipliers from K such that e 1.l. . = v..b.. (4.10) IJ I J . We illustrate the procedure, where k is 2, as the first step in an induction argument. In this case,

B

= (bll,

b12 > b212 b,,),

while the following multiplicative identities hold : bllbll = bll, b22b22 = b22 > bllb12 = b12, b22 b21 = b21, b,,b2, = b22 bll = b1,b21 = b2, b12 = bI2bll = b,,b,, = 0.

In addition, we know that b,, b,, = crb,,, where CI E K. Denote ( l / ~ ) b ,by ~ b;, and observe that b,, b;, = bll. Moreover, the equality (4.1 1)

bilbl2 = Xb22

implies that Xb12 = X(b12 b22) = bl,(Xb22) = blA%Ibl2) = (b12b;l)bl2 = bllb12 = b,2 *

Consequently, in Eq. (4.1 l),

x = 1, so that (4.12)

b;, b12 = b22.

Now define the set {e,j), 1 5 i, j I 2, by ell =bll,

e2, = b 2 , ,

e21 =b;,,

e12 = b 1 2

to obtain the set M . Proceed by induction, assuming the construction for the case of k idempotents, 1 I kI n - 1. Consider the decomposition of J into the direct sum

J

= L,

+

+ L,

of the minimal left ideals {Li) with associated idempotents (e,), I I i 5 n. There exists a basis B

= {bij},

1I i,j 5 n,

4. The Simple Components of the Group Algebra KG

109

of J. By the induction hypothesis, the set B' = ( b i j } ,

1

i , j < n - 1,

can be scaled to obtain a set 1 I i , j 5 n - 1, M' = {eij}, having the required multiplicative properties. We wish to scale the elements of the nth row and nth column in order to obtain a larger set

M = (eij>,

(4.13)

1I i, j I n,

so that

e I.J. ers = 6 j.r e i.s for all admissible values of i, j , r, and s. First take en, to be the idempotent generator e, of L, determined in (4.1). Then scale b,, to obtain elements el, = b,n

such that

and

enleln= e n ,

en1

z=

vnlbnl,

elnenl= e l l .

and

Next define elements

e,~ . = y .J n bJ.n ,

1
such that

ejleln= ejn, 1
1 < i < n,

such that

el,eni = e l , ,

Finally, define e, I . = v . b .

1 < i < n.

i < n, by means of these definitions, such Obtain elements eniand e , , 1 I that

(4.14)

1
ejleln = ej,,

elneni= e l , ,

1 I i 5 n.

It now follows that for, 1 5 j < n, 1 I k < n, (4.15)

ejkekn

= ejk(ekleln) = (ejkekl)eln = ejleln = e j n .

Furthermore, for 1 5 k I n, there exists a constant enk ekn

Consequently, Xk

= Xk

=X k

=

*

ekn)

so that xk is 1 for each k. This implies that en, = enkekn,

xk such that

= elk

ekn

1I k I n.

=

7

110

2. The Representation Theory of Finite Groups

In summary, for 1 ~j i n and 1 I k I n,

ejkekn= e j , ,

(4.16)

ejkern= 0,

k#r .

This result extends the construction from the case of n - 1 idempotents to that of n, completing the induction. It follows that there exists a basis M with the multiplicative properties defined by (4.16) for each minimal twosided ideal J in the decomposition of the group algebra KG. Such a basis is referred to as a set of matrix units for J. We introduce a basis { E i j } , 1 i i , j 5 n, of the n x n matrices over the field K of complex numbers, such that the matrix E,, has exactly one nonzero entry, which is 1 at the intersection of the rth row and the 8th column. Let the element in the ith row and j t h column of the matrix A be denoted by a i j . Then A has the unique expansion A

(4.17)

= i,

i

uiiEij.

For the case where n is 2, the explicit expression for the elements of this basis are

A linear transformation T from a finite-dimensional K-space U to a finitedimensional K-space V is completely determined by specifying the images {T(u,)}of a basis {uJ, 1 I i i n, of U. This theorem can be applied, in particular, to the basis {eij}, 1 I i, j I n, of a two-sided ideal J and the basis { E i j } , 1 i i, j i n, of the space Kn. We denote by T the unique linear transformation from J to Kn defined by

T(eij)= E i j ,

(4. IS)

1 i i, j I n.

It follows from the definition of T that T(ejkek,) =

T(ejn)= Ejn = Ejk Ekn = T(t?jk)T(ek,).

Also, if k # r, then

T(ej,ern)= T(0)= 0 = EjkErn= T(eik)T(er,). Therefore, T is a linear transformation of J onto Kn preserving the multiplication of the basis elements. Let

111

4. The Simple Components of ?he Group Algebra KG

Then

C i , j m1 , n tijqmnEijEmn

Hence, Tis a linear transformation of J onto Kn which preserves the operation of multiplication of vectors, that is, T is a homomorphism of J onto K,. However, T is an isomorphism of the vector space J onto the vector space Kn. Consequently, T is an isomorphism of the algebra J onto the algebra Kn of all complex n x n matrices. The extension of this result to the group algebra KG of a finite group G over the field F of complex numbers is now merely a matter of introducing a suitable notation. We use the symbol

A

= A ( l ) 0 . .*

0 A(r)

for the set of all quasi-diagonal matrices of the form

iI r. where each A(i) denotes the algebra of all ni x ni complex matrices, 1 I The set A is a subalgebra of the algebra K, of all cx x ci complex matrices where u = n,

+ + n,.

Let the u x u matrix E j k iconsist entirely of zeros except in the A(i)th block where it has the single nonzero entry 1 at the intersection of the j t h row and kth column. The set of all matrices {Ejki},1 Ij,k I ni, 1I i 5 r, constitutes a basis of the matrix algebra A. Suppose that the group algebra KG of a finite group G over the field of complex numbers has the decomposition KG

= J1 @

0 J'

2. The Rrpresrntution Theory of Finite Groups

112

as the direct sum of simple two-sided ideals, and, furthermore, that each J' has the decomposition

J'=L,'@**.@L,,,'

as the direct sum of minimal left ideals {Lj'}, 1 < j 5 n,, 1 5 i 5 r. According to the preceding discussion, there exists a family (Ti), 1 5 i 5 r, such that each Tiis an isomorphism of the two-sided ideal J' onto the algebra Kni of all n , x n i complex matrices. Every x of KG has the unique decomposition

x

= x1

+

9

.

.

+

XI,

x iE J',

15i

r.

We define a linear transformation T of KG onto A by means of (4.19)

T(x)= T,(x')+ . . . + T,.(x').

This linear transformation T has the properties that (4.18')

T(ejki)= Ejki

and (4.20)

T(ej,'e,,') = EjkiEs,'.

Consequently, T is a K-isomorphism of KG onto A which preserves the multiplicative operation for a basis of KG. It follows that Tis an isomorphism of the algebra KG onto the matrix algebra A . We phrase this final result in the form of the following theorem. (4.21) THEOREM. The group algebra KG of a finite group G over the field K of complex numbers is isomorphic to an algebra A of complex matrices, each of which is a matrix in the same quasi-diagonal form. The number of blocks in the quasi-diagonal form of the matrices of A is equal to the number of simple components of the semisimple algebra KG. The basis {ejk'].1 <,j, k 5 n , , 1 i 5 r, is called a symmetry basis of KG or a s~w7nwtuyadapted basis, since it reveals the symmetry of the group C as reflected in its distinct classes of equivalent, irreducible representations. On the other hand, the basis { g * } , g E G, is called the natural basis of KG. A general solution of the problem of relating effectively the symmetry basis to the natural basis of an arbitrary finite group G has not been obtained. Methods are known for certain special groups, for example, the symmetric group. However, since the results for the symmetric group are too complicated to present in a short space, we turn to other considerations. (4.22) THEOREM. A linear representation T of the group algebra KG on the complex representation space V is faithful if and only if the decomposition of V into irreducible subspaces of T contains a subspace KG-isomorphic to each minimal left ideal of KG.

113

4, The Simple Components of the Group AJgebra KG

Proof. It follows by Theorem (2.33) that if the direct sum decomposition

v = v, 0

*

* . 0 v,

of the representation space V into irreducible subspaces fails to contain a subspace that is KG-isomorphic to some minimal left ideal L of KG, then T is unfaithful. Conversely, suppose that any given minimal left ideal L of KG is KG-isomorphic to some ireducible subspace, say W, of V. Then there exists a KG-isomorphism h of L onto W such that the vector

h(ax) = ah(x) = T(a)h(x),

a E KG, x E L

is zero if and only if the vector ax is zero. If the vector ax is zero for every choice of the minimal left ideal L, then a is zero. Therefore, T is a faithful representation of KG. (4.23) THEOREM. A linear representation T of a finite group G on a complex vector space V of dimension n is irreducible if and only if lm T contains a linearly independent set of n2 linear transformations. Proof. Note that T is an irreducible representation of C if and only if T* is an irreducible representation of the group algebra KG. However, T* is irreducible if and only if V is KC-isomorphic to some minimal left ideal L of KG, that is, if and only if Im T* is isomorphic to K, where n is the dimension of either L or V over K. Since Im T is a generating set of Im T*, it follows that V is irreducible if and only if Im T contains n2 linearly independent linear transformations where n is the dimension of V. (4.24) THEOREM. If T is an irreducible linear representation of a finite group C with an n-dimensional, complex representation space V, then T occurs n times in the regular representation R. Proof. According to Theorem (2.31), every irreducible representation T of the group algebra KG is equivalent to some representation !RL where L is a minimal left ideal of KG. Such a minimal left ideal L of dimension n is contained in a simple, two-sided ideal J which is the direct sum of n minimal left ideals, each of which is KG-isomorphic to L and hence to V. Furthermore, every minimal left ideal L’ equivalent to L is contained in J. This establishes the theorem. The center o f a group G is the subset of G consisting of all elements which commute with every element of G. The center is denoted by C ( C ) ,that is, (4.25)

C(G)

= { z :z E

G, zx

= xz

for x E Cj.

The center of an algebra A is the set, (4.26)

C(A)

= {z: z E

A, zx

= XZ,

x E A}.

114

2. The Representation Theory of Finite Groups

An element z of the center of KG has the form z=

For any g

E

(4.27)

G, one has

c z(t)t 9-1 c z(t)gtg-' c z(g-'tg)t.

z =gzg-1 = g =

c z(t)t.

tcG

(teG

=

teG

tcG

This shows that when z E C(KG), z(g-'tg)

=~

t, g

(t),

E

G.

In different words, if z belongs to the center of KG, then z has the same value for any two conjugate elements of G. The converse is also seen to be true. If an element z has the same value for every conjugate pair of G, then z belongs to the center. The reason is that the equation, g , t E G,

z(gtg-'> = z(t>,

implies that gzg-' = z ,

gEG,

or

gz=zg,

gEG.

Since every element in KG is a linear combination of elements of G, it follows that xz = Z X , XEKG. Thus z belongs to the center of KG. Denote the set of distinct classes of conjugate elements of G by { K l ,. . . , K,.} where the ith class, (4.28)

Kz

= (gl i 5

..

* >

ghi.

il,

contains hi elements. The set of elements zj=g,j+*..+g,,,,j,

I
constitutes a set of generators for the center, Since this set is linearly independent, it forms a basis of r elements of C(KG). Next we prove that the dimension of the center coincides with the number of simple, two-sided ideals in the decomposition KG

(4.29)

= J' @

. . . @ J',

of the group algebra KG over the field K of complex numbers. Let x

=

C tik'eik'+ ... + C t i l e i :

i, k

i. k

5. Introduction to Group Characters

115

be an element of the center. For every permissible choice of u, u, and w, the equation

euw"x = xe,," is valid. Consequently, (4.30)

C tw/eo; k

C
i, k i,

i

Now, since the set of the e,: is linearly independent, it follows that all of the previous coefficients must vanish except the pair 5,," and to,"which are equal. It follows that the part xu of x which lies in J" must be of the form

+ + 5,1"enu,n, = 511"eu,

xu =
where e" is the idempotent generator of J". Thus, x in the center implies that (4.31)

x = tle'

+ ... + <,e',

where each e", 1 5 u r, is the idempotent generator of the corresponding simple ideal J". On the other hand, every x of the form of Eq. (4.31) is an element of the center. Hence the set {el, . . . , e'} is a basis of the center. It follows that the number of distinct classes of conjugate elements in G coincides with the number of simple ideals in the decomposition given by Eq. (3.30), which in turn coincides with the number of distinct classes of equivalent irreducible representations of the finite group G. We turn now to another aspect of representation theory which depends upon a new concept, that of the character of a representation. By means of the theory of characters we shall be able to associate with every representation of the finite group G over the field K of complex numbers a unique set of numbers which characterizes the representation. 5. INTRODUCTION TO GROUP CHARACTERS

The concept of the character aflorded by a linear representation T of a finite group G with the representation space V over the field K of complex numbers is defined in Section 5. Orthogonality relations are developed for the characters of such irreducible representations of G . Two representations T and T of G are proved to be equivalent if and only if they afford the same character. A method of deciding the irreducibility of a representation T from the character it affords is described. Illustrations of the basic theorems are obtained by examining the character tables of several groups of low orders. The center C(KG) of the group algebra KG of a finite group G is investigated

2. The Representafion Theory of Finite Groups

116

and the structure constants of C(KG)are introduced. The use of the structure constants in determining the irreducible representations of G is illustrated. Let T be a linear representation of the finite group G on the n-dimensional representation space V over the field K of complex numbers. The character afforded by the representation T is the mapping x from G to the complex numbers such that (5.1)

x(g)

= tr[T(g)l,

9 E G.

Remember that the trace of a transformation T(g) belonging to Hom,(V, V) is the trace of the matrix of T(g) with respect to any basis. It is a familiar fact that this trace is independent of the choice of the basis of V. Also, if A and B are n x n matrices, then A B and BA have the same trace. (5.2) THEOREM. If Ti and T , are equivalent linear representations of the finite group G with representation spaces V, and V,, respectively, then TI and T2 have the same character. Proof. By assumption, there exists an isomorphism h in Hom,(V,, V,) such that g E G. hT1(g)K1 = T,(g), However, it is known from vector space theory that trt~,(g)I= tr[h-lT2(s)hI

= trtT2(g)hh-'I = trtT2(g)I,

which implies XYS)

= x2(9),

9 E G.

The following proof is a variation of that of Theorem (5.2). Let T be any representation of G and let g' and g be conjugate elements of G, that is, let there exist an x E G such that g' = xgx-1.

If

is the character of T, then z(g')

= tr[T(g')] = tr[T(xgx-'>] = tr[T(x)T(g)T(x-')I = tr[T(g)l = x(g).

It follows that the character x of a representation T is a class function on G, that is, a function f whose value at x coincides with its value at y if x is conjugate t o y .

(5.3) THEOREM. Let T be a decomposable linear representation of the finite group G with complex representation space U. Let U be the direct sum V @ W of the reducing subspaces V and W on which T defines the representations T , and T, , respectively. The character x of T is the sum of the characters and x2 of TI and T,, respectively.

117

5. Introduction to Group Characters

Proof. Let the sets {v,, . . . ,vm} and {wl,. . . , wn} be bases of the subspaces V and W, respectively. Their union {vl, . . . , v,, ; w,, . . . , w,,} is a basis of W. Also T(g)v, = Tl(g)vi = uiivi

+ vi',

and T(g)wj = T2(g)wj = pjjwj

+ wj',

1ii I m, 1 < j 2 n,

where the expansions of vi' and wj' in terms of the given bases have no diagonal terms. Consequently,

xk)= @ I + . . . + a m 3 + ( P I I + . . . + b,") = x1(9) + Z2(9)> 1

as was to be shown. Let { K , , ..., K,} denote the distinct classes of conjugate elements of the finite group G. If x and y are elements of K i ,then

x(x> = x(v>

for any character x. This common value of x for elements of the class Ki is denoted by the symbol xi. Since the number of distinct classes of conjugate elements of G is equal to the number of distinct classes of equivalent complex irreducible representations, the characters of these distinct classes of equivalent irreducible representations can be labeled

x1

2

... >xr.

The standard symbols for the values of the character xi of the ith class of irreducible representations are x1

i

i

,... ,XI.,

where xii denotes the value of the character xi for the members of the class K j of conjugate elements, I I j I r. A complete presentation of the characters of the distinct classes of equivalent irreducible representations of the finite group G is made by means of an r x r table, where r is the number of distinct classes of conjugate elements of G. It is worth remarking that this result of a square character table is not necessarily d i d in the case of certain representations over special fields. In the following two theorems, we derive orthogonality relations among the rows and columns of the character tables of irreducible representations of finite groups over the-field of complex nunzbers.

(5.4) THEOREM. Let T and S be nonequivalent, irreducible linear representations of the finite group G with complex representations spaces U and V. Let {u,}, 1 I i I nz, and {vi}, 1 I iI n , be bases of U and V, respectively. For g E G, let {bji(g)} and {uji(g)} be the matrices of T(g)and S(g) with respect

I18

2. The Representation Theory of Finite Groups

to the given bases. The following equality holds for all permissible values of i. j , k , and r .

c ajk(g)br'(g

- 1)

= 0.

qsG

Proof. By assumption, for g E G, S(g) belongs to GL(V) and T(g) belongs to GL(U). Let C be any element of Hom,(U, V) a n d denote by P the element of Hom,(U, V) defined by

P

(5.6)

=

1S(g)cT(g-').

qEG

For each h E G,

=

c S(q)CT(g

qtG

-117)

=

qeG

S(g)cT(g- ')T(I7)

= PT(11).

From Schur's lemma, it follows that P must be the null homomorphism of U into V. We introduce a family C(i,,j), 1 5 i 2 rn, 1 2.i 2 n , of linear transformations of Hom,(U, V). Each linear transformation C(i,j) is defined by (5.8)

C(i, j)u, = v j

and

C(i,j)uk= 0,

i # k.

Select a n arbitrary member C ( i , j ) of this set to define a particular P by Eq. (5.6). Then it follows that

Since the set { v ~ }1. 5 k I n , is linearly independent, it follows that

(5.9)

c a,"g)b,'(g-')

=0

yf=G

for all permissible choices of i, j , k , and r, as was t o be shown.

(5.10) THEORIYM. Let S be an irreducible linear representation of the finite group G on the n-dimensional complex representation space U with basis {ui). 15 i < 1 1 . If ( a j ' ( g ) }denotes the matrix of S(g), the following equation holds (5.11)

119

5. Introduction to Group Characters

Proof. Let C ( i , j )denote the element of Hom,(U, U) defined by

C(i,j)u;= u j

Since

p

C(i,j)ur

and =

i # r.

= 0,

c %)c(m%7-1)

9eG

commutes with each element S(go),g o E G, by the second form of Schur's lemma,

where 1, denotes the identity operator on U. Consequently, (5.12) ciju, =

9eG

S(g)C(i,j)S(g-')u, =

2 S(g)C(i,j) 21 a,'(g-')u,

9eG

t=

It follows from the linear independence of the set {ui}, 1 5 i 5 n, that

c ajf(g)as'(g-

9 EG

I)

= askij.

Take t equal to s and sum on the index s so that

c i:a,yg

g€G

s=1

-')aj"(g) = ncjj .

Hence

(5.13)

so that

cij = Sji{[G: 11,'~). Finally,

(5.14)

1 aj'(g)a,'(g-')

gtG

= d,'dj'[G : l]/n,

which completes the argument. Denote by xu and xu the characters of the irreducible representations S and T of Theorem (5.4). Consider the special case of Eq. (5.5) obtained by taking r equal to i, j equal to k so that (5.15)

1 a , k ( g ) b , ' ( f ' )= 0.

9s G

120

2. The Representation Theory of Finite Groups

By summing on k and i i n Eq. (5.15), one obtains

c x"(s>x"(s '> -

gtC

= 0,

whenever x" and xu are characters of nonequivalent irreducible representations of the finite group G. A similar summing, using the results of Theorem (5.10), yields the result that

(5.16) where x" is the character of an irreducible linear representation S of the finite group G. These results may be summarized in a single equation

(5.17) where xu and x" are characters of the irreducible representations T" and T" of the finite group G. A linear representation T of a finite group G on an inner-product space V is said to be a unitary representatiot? if and only if T(g) is a unitary transformation for every g E G. When T is a unitary representation, it follows that T(g-')

=

[m)l-'= [Ug)l*.

Since the matrices of T(g) and [T(g)]* with respect to a unitary basis of V are conjugate transposes, one has (5.18)

x(s-I) = tr[T(g-')l

= tr[(T(g))*l = X(g).

There is a theorem, not established in this book, that every linear representation T of a finitc group G with a finite-dimensional representation space V over the field of complex numbers is equivalent to a unitary representation. Consequently, (5.18') for all such representations. This result implies that Eq. (5.17) can be written in the form (5.19)

c x"(g)j"(g)

9EG

= 6,"[G : 11.

Since each conjugacy class Ki contains h i elements for which the values of the character coincide, Eq. (5.19) implies that (5.20)

121

5 . Introduction to Group Characters

The character table of a finite group G with r distinct classes of conjugate elements xi1

*-.

x2l

x12

~2~

xlr

xzr

*..

* * .

xrl

xr2

xi

can be regarded as an r x r matrix C with elements (xji>.Let D be the matrix whose elements {dj'}are given by d.' = h i X j .

By Eq. (5.20), D is equal to [G : l ]C-l , that is, C D = [G : l ] I r , where I, is the r x r identity matrix. This means, of course, that

D C = [G : 111,. a result which can be stated in the form r

(5.21)

or (5.21')

r

. .

11 h,x,'j,J

j=

= 6,"[G

: 11.

The complete reducibility of a representation T of a finite group G over the field K of complex numbers together with Eq. (5.17) permit the identification of the equivalence class of such a representation T merely from the character it affords. To be specific, by means of Theorem (1.45), T can be written uniquely as (5.22)

T = a, TI 0 * . . @ a, T',

where the set { T I , . . . , T') contains exactly one representative from each of the r distinct classes of equivalent irreducible representations of G while the integers {u1,. . ., a,) indicate the number of times each representation occurs in the decomposition (5.22). The representation T is equivalent to any other representation T' in which these irreducible summands appear the same number of times. Moreover, the numbers (a,} can be determined from the character x of T i n the following manner. From the natural extension of Theorem (5.3), the character of T is given by (5.23)

,

x = alxl + . . . + arxr = C aj xj. j = 1

122

2. The Representation Theory of Finite Groups

Form the expression, (5.24)

c X(S)%"(S) c c ajxj(s)x"(s) =

SEG

gtC

= a,[G : 11.

j=1

Thus we find that (5.25) Consequently, the number of times that the irreducible representation T" occurs in the representation T is completely determined by the character of T. This result together with Theorem (5.2) leads to the following theorem. (5.26) THEOREM. Two linear representations T and T' of the finite group G over the field of complex numbers are equivalent if and only if the characters x and x' of the respective representations coincide. We continue with the observation that Eqs. (5.23) and (5.24) imply (5.27) Thus, in conjunction with Theorem (5.26), one obtains the following theorem. A linear representation T of the finite group G over the (5.28) THEOREM. field K of complex numbers is irreducible whenever the right-hand side of Eq. (5.27) is equal to [C : 11 and reducible otherwise. Equations (5.24) and (5.27) can be written

C hi xi j i u = a,[G : 11,

i= 1

(5.29)

i, l ? i ) ( i j i

i= 1

=

[G: l](a,2 + . . . + a;).

Now let T be a linear representation of the finite group G on the n-dimensional representation space V over the field K of complex numbers. We summarize a number of facts about T and its character x. The image T(l)of the group identity 1 of G is the identity 1" of Hom,(V, V) so that

~ ( 1= ) x, = tr[l,]

(5.30)

= n.

Consequently, the first column of the character table X lists the dimensions r distinct classes of equivalent irreducible representations of G. By convention, the first class K , of conjugate elements of a finite group G consicts of the identity alone so that h , is 1. Thus, by Eq. (5.21'),

i n , . . . . , n,) of the

(5.31)

n12

+ . . . + nr2 = (x,')' + . . . + (x,~)' = [ G : 11,

a result previously noted. Let 91 denote the regular representation of a finite group G. The equation %(s)t = st, S, t E G,

123

5. Introduction to Group Characters

implies that the matrix of %(s) with respect to the natural basis has no diagonal entries, except in the case of the identity element of G. Therefore, if x denotes the character of the regular representation, then

(5.32)

X(S) =

0, s # 1,

~ ( 1= ) [ G : 11.

Applying the first of Eqs. (5.29) to the regular representation, one obtains

(5.33)

a,[G : 13 =

1 h i x i j i U= h , X , j I u= n,[G : 13,

i= 1

so that the number a, of copies of the irreducible representation T" in the regular representation % is equal to the dimension of the representation T", a result obtained in Section 4.

(5.34) EXAMPLES. One must have available either some examples of irreducible characters or elementary methods of determining them in order to illustrate the application of these results to the calculation of character tables. To this end, note that every finite group G has a class of irreducible representations, each of which makes every element of G correspond to the identity transformation 1 of some one-dimensional vector space V. This class of irreducible representations is called the 1 -representation and the character it affords the 1-character orprincipal character of G. The 1-character, of course, makes every element of G correspond to the number 1. Each irreducible representation T of a factor group G / H of G modulo one of its normal subgroups H determines .an irreducible representation T of G. More generally, let h be a homomorphism of G onto G'. Then every irreducible representation T' of G' with representation space V determines an irreducible representation T of G since the composition T' h is a homomorphism T of G into GL(V). An interesting special case arises when H is a subgroup of index two of the finite group G. Then G / H is a cyclic group of order two generated by some left coset g H of H . Consequently, G / H has two distinct classes of equivalent irreducible representations '%I and '3' where 93' makes gH correspond to 1 and 93' makes it correspond to - 1. These give rise to two irreducible representations of G, 0

T'

= %'

o

v

and

T 2 = 912 v, 0

where v denotes the natural homomorphism of G onto G/H. T' is merely the I-representation back again. On the other hand, T 2 is an irreducible representation said to belong to the normal subgroup H . It is also called an alternating representation since its character takes only the two values, 1 for elements of H and - 1 for elements of the other left coset of H . The most famous example is that of the alternating subgroup A, of the symmetric group S , . Not uncommonly, a group G may have several different subgroups of index two, each of which gives rise to an alternating representation. This is the case of C , @ C , which is discussed below. The method is also

124

2. The Representation Theory of Finite Groups

useful when the factor group G / H is cyclic or, more generally, abelian. Naturally, whenever the irreducible representations of the factor group G / H are known, the structure of G / H is a matter of indifference. We begin with the application of these ideas to the group S , whose Cayley table is shown in Table (5.35). The group S , has three distinct classes of (5.35)

CAYLEY TABLEOF S3 123456 1123456 2231645 3312564 4456123 5564312 6645231

conjugate elements: K , = {1} with 11, = 1 , K, = (2, 3) with h, = 2, and K, = {4, 5 , 6) with / I ,= 3. Consequently, the number of distinct classes of equivalent irreducible representations is three so that the character table of S , is a 3 x 3 matrix. Since the subset (1, 2, 3) is a subgroup H of index two of the group S , , the I-character and the alternating character provide the first t w o rows 1 1 1 1 1 -1 of the character table, leaving a third row x, y , z to be determined. Equation (5.31) implies that 1’ + 1’ + X’ = 6, so that x has the value 2. Equation (5.21) then gives l(1)

+ l(1) + 2y = 0

and

l(1)

+ 1(-1) + 22 = 0.

It follows that the character table of S , has the form of Table (5.36). (5.36)

CHARACTER TABLEOF S3

The character table of C,@ C, is needed to apply these methods to the quaternion group Q, one of the two distinct classes of isomorphic, nonabelian groups of order eight. The Cayley table of C, @ C,, sometimes

125

5. Introduction to Group Characters

called the Viergruppe, is shown in Table (5.37). Each element other than 1

(5.37)

CAYLEY

TABLE OF

cz 0cz

1234 11234 22143 33412 44321

has order two, so that C, @ C2 contains three distinct normal subgroups of index two, namely,

H2 = (1, 2},

H , = { I , 3},

and

H4 = { I , 4).

In addition to the 1-representation, there are three different alternating representations defined by the three normal subgroups. Denote the characters of these representations by x 2 , x3, and x4 respectively. Since C, @ C, has four classes of conjugate elements, it follows that these four characters constitute a full set for the distinct classes of equivalent irreducible representations. The character table of the group is shown in Table (5.38). With this

(5.38)

CHARACTER

TABLEOF cz 9c z

KI Kz K3 & 1 1 1 x 2 1 1 - 1 -1 x 3 1 -1 1-1 x4 1 - 1 -1 1 X I 1

table available, we turn to the analysis of the quarternion group Q whose Cayley table is shown in Table (5.39).

(5.39)

CAYLEY TABLE OF

THE

QUATERNION GROUPQ

12345678 112345678 223416185 334127856 441238567 558763214 665874321 776581432 887652143

126

2. The Representation Theory of Finite Groups

According to its Cayley table, the group Q contains a subset ( 1 , 3} constituting a normal subgroup H whose left cosets in G are

H

= {I,

3}, 2H = {2,4], 5 H

= (5,

7}, and

6H

= (6, S}.

The Cayley Table of Q / H is shown in Table (5.40).

(5.40)

CAYLEY TABLEOF QIH 1H 1H 1H 2H 2H 5H 5H 6 H 6H

2H 2H 1H 6H 5H

5H 5H 6H 1H 2H

6H 6H 5H 2H IH

A comparison of (5.40) with (5.37) shows that the mapping f from the factor group Q / H to the Viergruppe defined by f { l , 3) = 1 , f(2, 4) = 2, f { 5 , 7) = 3, and f(6, 8) = 4 is an isomorphism. If S',%', !R3, and 914 are nonequivalent irreducible representations of the Viergruppe and v is the natural homomorphism of Q onto Q / H , then the compositions

T'

=

s'

of

3

v , T 2 = '%*

of o

v, T 3 = S 3 o fa v ,

and

T4= S4o f a v,

are nonequivalent irreducible representations of the quaternion group Q. These define corresponding characters x', x', x3, and x4. The character table of the quaternion group Q is shown in Table (5.41), where the entries for (5.41)

CHARACTER TABLE GROUPQ QUATERNION 1 2 3 1 1 x ' l x2 1 1 1 x3 1-1 1x4 1 - 1 1x 5 2 0--2

4 5 6 7 8 1 1 1 1 1 1 - 1 -1 -1 -1 1 1-1 1-1 1 -1 1-1 1 0 0 0 0 0

characters xl,. . . , x4 were made from the Table (5.38) while those of xs were determined afterwards by (5.3 1) and the orthogonality relations. The character table gives the value of x i for each element of the group Q rather than merely for the classes, a more lengthy presentation convenient for some purposes. This completes the discussion of the most basic methods of finding the character tables of a small group. The next procedure to be discussed is applicable in theory to the computation of the character table of any finite group. Its practical application is

127

5. Introduction to Group Characters

limited by the complicated matrix calculations involved. To begin the discussion, we recall that a subset B of an algebra A is a subalgebra of A if B satisfies the axioms of an algebra under the operations and with the field of A. In other words, B is a subspace of the vector space A such that xy E B whenever x, y E B. Although an ideal J of an algebra A is necessarily a subalgebra, a subalgebra B of A is not necessarily an ideal. (5.42) EXAMPLE. The set B of all matrices of the form

I1 a", I

is a subalgebra of the algebra A of all complex 2 x 2 matrices. Nevertheless, B is not an ideal of A, since neither

nor

belong to B whenever a,, is different from zero. We observed in the discussion preceding Eq. (4.29) that the set of elements

(5.43)

zj=glJ+*-.+gh,,J, 1< j < r ,

is a basis of the center C(KG), a subspace of KG. Although the center C(KG) is not an ideal of KG, it is easy to see that the product xy E C(KG) whenever x, y E C(KG) so that C(KG) is a subalgebra of KG. In particular, the product z,z, of two basis elements is a linear combination (5.44)

z,z, = C,"1Z,

+ . . . + c,,,z,

of the basis elements. Since each factor of the product is a linear combination of g i , 1 5 i < [G : 13, with nonnegative integer coefficients, it follows that each such g i appears on the right of (5.44) with a nonnegative integer coefficient and, consequently, so must each z k . The set (5.45)

{c,,,},

1 < u, u, u' 5 r,

of coefficients consists of nonnegative integers which are called the structure constants of the algebra C(KG). To relate these results to earlier ones, recaIl that the group algebra KG decomposes into the direct sum (5.46)

KG=J'@..*@J'

128

2. The Representation Theory of Finite Groups

of its simple components in essentially one way. Each class of irreducible representations of KG contains a member !RL which is a representation induced by the regular representation 93 on some minimal left ideal L contained in the simple component J" of KG. The ideal L is equivalent to an ideal L," which occurs as a summand in the decomposition of J" according to the symmetry basis {e","}, 1 5 u, w I n u , 1 I u 5 r, of KG. The representation 91L,,,,,belongs to the same class of irreducible representations as SL. We are able to obtain the matrix representation afforded by 'iRLWUwith little difficulty. Note that if (5.47)

=

C tj,"ej,", j

1 Iv 5 nu.

Therefore, the matrix representation M afforded by %,+, x of KG correspond to (5.49)

M(x) = { t j / } ,

makes the element

1 Ij , u I nu.

The elements of M ( x ) are the components of x with respect to the symmetry basis of J". Thus every class of irreducible representations contains a member !XLV,
(5.50)

ei = e l l i

+ . . . + eninii,

1I iI r,

also form a basis of the center of KG. Consequently, there exist matrices {aj'} and { x j i ) such that (5.51)

ei =

(5.52)

z, =

1 ccj'zj, I

.

j = 1

r

C w,,jej,

j= 1

I ii s r ,

1 i u I r.

The component of the element z, in the ideal J k is (5.53)

z,ek = w,kek,

129

5. Introduction to Group Characters

as can be seen from (5.52). The image of z, in the matrix representation M determined by Jk is the matrix w,kInk

5

where In,denotes the nk x nk identity matrix. It follows that the character xk has the value (5.54) where

x"zJ

k

= x1 w ,

=

k

3

xlkis n k , the degree of the matrix representation

(5.55)

XIkw,"

= xk(zu>= Xk(glu

+ . + g h , , u) * *

M . Finally,

= huXk(g),

where g belongs to the class K, . Thus one obtains (5.56)

x,"

= Xk(S) = xlkw,k/hu.

We observed in passing that

z,ek = w ,kek , which means that the left-translation zuL,determined by z,, has ek as an eigenvector with the corresponding eigenvalue ~ ~ ( z , ) / n ,The . eigenvalues and eigenvectors of zULcan be computed from its matrix {c",,,,,}, 1 5 u, w 5 r, of structure constants with respect t o the basis {zl, . . . , z,}. When a given eigenvalue w," has a one-dimensional eigenspace U,", an eigenvector Y corresponding to w," establishes by (5.53) and (5.56) the kth row of the character table up to a scale factor. This factor is determined by the orthogonality relations for the rows of the character table. Such an immediate solution is frequently thwarted by the fact that some of the eigenspaces of any particular zUL have dimension greater than one. Let the left transIation zULcorresponding to

(5.57)

z,

= Ute'

+ + wie' * *

have an eigenspace U of dimension s greater than one. Then U has a basis {eil,.. ., eis} of central idempotents, each of which appears in the expression (5.57) with the same coefficient mui, which is the eigenvalue corresponding to U. It is clear that the left translation zL defined by most linear combinations of the form

(5.57')

z=alel +-**+are'

does not have repeated eigenvalues. This implies, of course, that most left translations zL defined by linear combinations of the form (5.57")

z = blz,

+

'

*.

+ b'z,

130

2. The Representation Theory of Finite Groups

do not have repeated eigenvalues. The eigenvalues of the left translation zL defined by (5.57") are given by w 1 = b 'o , '

+ ... + b'o;

or= blw*'

+ . . . + b'w,'.

(5.58)

The matrix { w j i }of the coefficients on the right-hand side of (5.58) is . nonsingular since it represents a change of basis. It follows that a suitable choice of { b ' , . . . , b') will lead to any prescribed set of values of {o',. . ., or} of the eigenvalues of z L . As a practical matter, one can take linear combinations of the matrices {c,,,), 1 5 v, WJ i r, 1 5 u 5 r, in order to obtain a matrix with distinct eigenvalues whose eigenvectors determine, up to a scale factor, the central idempotents { e l , . . ., er} and consequently the character table of the group. Another approach is to introduce a set of indeterminates { t ' , . . . , t'} in order to determine a family {zL(t', . . . , t')) of linear transformations which satisfy the eigenvalue equations

(5.59)

z , ( t ' , . . . , tr)ek=

(

C tlZi i

.

1

ek =

C t'o? ek = w ( t ' , . . ., t')ek.

( i

To develop the component form of (5.59), one expands ek in terms of the basis { z n I ]1, I nz 5 r , obtaining

By comparison of coefficients of z,,, , one finds (5.61) or

(5.62) The deterrninantal function, (5.633

( 6 , ' ~-

1 ticijmI =f(w;t ' , . . . , tr), I

5. Introduction to Group Characters

131

is a homogeneous polynomial in the r + 1 variables. w, {ti}, 1 I i 2 r. For suitable specializations {b', .. . , b'} of the indeterminates {t', .. ., t'}, the set of equations (5.62) has nontrivial solutions, namely, the coefficients of the set of vectors {ek}, 1 5 k S r, with respect to the basis {z"}, 1 s u s r , of the center C(KG). Furthermore, for these suitable specializations, the corresponding roots (mi}, 1 5 i S r, off(w) are distinct and given by (5.58). Thus one finds, in such instances, that (5.64)

f(w)= (O - a'). (O - o r )= [W - (blw,' + . . . + b'w,')] x [o- (b'o,2 . . . + b'o,2)]. * . [w - (b'w,' + . . . + b'o;)]. '

4

+

From continuity, it follows that this factorization is valid for an infinite number of distinct choices of each of the b', 1 2 i 2 r. Consequently, the factorization given by (5.64) is valid when the set {b'} of constants is replaced by the set {ti) of indeterminates. Finally, (5.65) f(w,t',

. . . ,t') = I d , , , j ~- 1 t i c j i m ( i

+ ' . + t'o,')]

= [O- (t'o,'

1

+ . . . + t'Q,2)] - . -

x [o- ( t ' o , 2

+ ... + t'o;)]

x [o- (t'w,'

is the complete factorization. To fix the ideas, we apply these considerations to the symmetric group S3 whose Cayley table is (5.35) and whose distinct classes of conjugate elements are K , = (11, K2 = (2, 31, and K , = (4, 5,6). The basis {zl,z 2 , z,) is defined by z1 = I, z2 = 2

(5.66)

+ 3,

and

23

=4

+ 5 + 6.

Note that (5.67) z2 2,

+ 3)(4 + 5 + 6) = 2(4) + 2(5) + 2(6) + 3(4) + 3(5) + 3(6) = (2

=6+4+5+5+6+4 = 22, = OZ, OZZ 22,

+

+

+

= ~ 2 3 1 ~ 1~ 2 3 ~2 2

+

c233z3.

Thus one finds that

c Z 3 ,= 0, Furthermore,

~ 2 3 = 2

0,

and

~ 2 3 = 3

2.

132

2. The Representation Theory of Finite Groups

since C(KG) is commutative. An easy computation gives the complete collection of structure constants, conveniently arranged,

(5.68)

These structure constants give rise to a determinantal function (5.69)

-t 3

-t2

f(w; t i , t2, t3) =

-2t3

w - t' - 2 t 2

-3t3

I

By adding the second and third rows to the first, one finds that (5.70) f ( w ; t', t2, t3)

+ 3t3)

w

- (tl

+ 2 t 2 + 3t3)

0 - (t'

0 - t' - 1 2

-3t3

+ 2 t 2 + 3t3)

-2t3

w - t' - 2 t 2

from which it follows that

(5.71) f ( w ; t ' ,

t 2 ,t 3 )

= [w - (t'

+ 2 t 2 + 3t3)]

1

-2t2 1 - 3 ~

= [w

-

(t'

+ 2 t 2 + 3t3)]

= [w

-

(t'

+ 2t2 + 3t3)][w - (t'

1 0 - t' -

-3t3

1

- t2)][w - (t'

+ t y 2 ) + r3(3), m2 = t ' ( 1 ) + t2(2) + w3 = t ' ( ~ + ) t 2 ( - 1) + t3(0). w' = t'(1)

Thus the matrix (wj'}, 1 (5.73)

t3(-3),

i, j

3, assumes the form

1

-2t3

0 - t l - 2t2

1

1

2t2 - 2t 3 w - t' - 2 t 2 3 t 3

The eigenvalues are given by the following expressions : (5.72)

t2

+

+ 2 t 2 - 3t3)].

I33

5. Introduction to Group Characters

In order t o compute the character matrix of S , , it is necessary to compute the values of xI1, x12, and x13, which equal the dimensions of the distinct classes of equivalent irreducible representations. From Eqs. (5.56), it follows that

c

(5.74)

hu

U

Xh,"= c I X I k I 2%%Lvu) U

or [G : 11 =

(5.75)

I xlk I

1w,k(Sij:/h,).

Finally,

I Xtk I

(5.76)

= tG : 11/[

By means of (5.76), we find that (5.77)

(xll)' ')IX(

U

c ~,k(W,k/hu)l.

+ 2(2/2) + 3(3/3)] = 1, = 6/[1(1/l) + 2(2/2) - 3(-3/3)] = 1,

= 6/[1(1/1)

(xI3)' = 6/[1(1/1) - I(-

1/2)] = 4.

Since their values are positive integers, one finds

xI1 = 1 ,

xI2 = 1, and xl 3 = 2. xI1, x12, and xI3, one can compute

Given these values of the character table by (5.56). For example, one obtains (5.78)

xll = 1(1/1) = 1 ,

xZ1= 1(2/2) = 1,

and

the entries of

x31 =

1(3/3) = 1.

Continuing this procedure, one finds the character table of S , to be as shown in Table (5.79). (5.79)

CHARACTER TABLE OF s 3

Table (5.79) agrees with Table (5.36) which was calculated by other methods. It is clear that practical use of this method depends upon the development of computer programs to carry out the details, otherwise the calcdation is too burdensome for any except small groups. An interesting by-product of this method is an expression for the central idempotents {el, . . ., er} in terms of the basis {zl, . . . , 2,) and the character

134

2. The Representation Theory of Finite Groups

table {zi'}, 1 I i, ,j I r . The matrix {oji}, 1 5 i, j 5 r , can be obtained imi, j I r, mediately from the character table. However, the matrix {aji}, 1 I such that

ei =

(5.51)

1 cxjizj

j =1

is the inverse of the matrix {wji}, more precisely, the transpose of (cxji> is the inverse of {coji}. The calculation can be done explicitly because of the orthogonality relations. Equation (5.52) can be written (5.80) by means of (5.56). This implies that (5.81)

1 j i z , = 1 :j

u= 1

u=l

=

Finally,

j=l

[G : 11

h,xi(e'/z,j)

=

c c /iux~j,"(ej/,ylj)

j=1 u=l

2 Gj(ej/xlj) = [G : 11 1 GJ(ej/X,j) = [G : l ] ( e k / ~ l k ) .

j= I

j =1

(5.82)

so that (5.83)

a;

= Xlkj,"/[G : 11.

This completes Chapter 2. Additional methods and results on representation theory can be found in Chapters 3 and 4 which are concerned with applications and examples. PROBLEMS

1. Let , j ' : G G' be a homomorphism of the cyclic group G = (x) generated by .Y. Show that f is completely determined by its valuef(x). --f

2.

Let j ' : G G' and 11: G' + G" be group homomorphisms. Show that is a group homomorphism. --f

/ I : , j ' : G + G"

3. Let G be the external direct product H @ K and let f :H + H' and h : K K' be group homomorphisms. Show that the m a p 2 : C + H' @ K' defined by i(x, y) = ( , f ( x ) h(y)) . is a group homomorphism. --f

Recall that the d e r i w d g r m p G' of a group G is the subgroup generated by all romnirtator,c .'i-'J--',YJ. formed from pairs {x,y} c G.

(a) Prove that the derived group G' is a normal subgroup of G. (b) Prove that the factor group G/G' is abelian.

4.

5. Let f : G - + A be a homomorphism of the group G into an abelian group A . Show that the derived group G' is contained in the kernel off.

135

Problem

6. Every one-dimensional complex representation T of a group G is basically a homomorphism of G into the multiplicative group K* of nonzero complex numbers. By Problem 5, the kernel of T contains the derived group G'. (a) Prove that T defines a representation of the factor group G/G'. (b) Prove that every irreducible representation of G/G' determines a one-dimensional representation of G. (c) Argue that all one-dimensional complex representations of G arise from those of G/G' in this manner.

7. The subgroup G' = {1,4, 5} is the derived group of the group G of order twelve with the following Cayley table. Using the ideas of Problem 6, find the one-dimensional complex representations of G. 1 2 3 4 5 6 7 8 9101112 2 1 4 3 6 5 8 710 91211 3 4 5 6 2 11211 7 8 910 4 3 6 5 1 2 1 1 1 2 8 710 9 5 6 2 1 4 310 91211 7 8 6 5 1 2 3 4 9101112 8 7 7 8 9101112 2 1 4 3 6 5 8 710 91211 1 2 3 4 5 6 9101112 8 7 5 6 2 1 4 3 10 9 1 2 1 1 7 8 6 5 1 2 3 4 1112 8 7 1 0 9 3 4 5 6 2 1 1211 7 8 9 1 0 4 3 6 5 1 2

8. Let S, denote the group of permutations on a set A = (a, b, c> [see (2.9)]. Let B = (va, v b , vc} denote a basis of the vector space V. For each g E S,, define T(g):V -+ V by T(g)v, = Show that T defines a representation of S , . 9. Let G denote a subgroup of GL(V).Suppose that W is a proper subspace of V such that g(W) c W for g E G. Show that the correspondence T : G + Hom(W, W), given by T(g) = g I W is a representation of G with representation space W. 10. Let / I : G G" be a homomorphism of the finite group G into the group G". Let x E G have order n. Show that h(x) has order dividing n. -+

11. Let T : G GL(V) be a complex representation of the finite group G. Let x E G. Show that the matrix of T(x) can be reduced to diagonal form, that is, T(x) is a semisimple linear transformation. -+

Look up the concept of a group being presented by generators and relations, for example, Rotman (1965) or Coxeter and Moser (1965).

2. The Representation Theory of Finite Groups

136

12. Let a group G be given by the generators {x,,. . . , x,) and the relations ym(x,,. , . , x,) = 1 (m = 1, . . . , t ) . Show that a map h : C + G from the group G into the group can be defined in such a way that h is a homomorphism iff the images { y , , . . . ,y,}, y i = h(xi), satisfy the relations y m ( y l , .. . ,y,) = i (m = 1, . . . , t ) . Note this result holds in particular for the case of representations. 13. The Cayley table of the symmetric group S , is given in (2.10). S , is generated by R, and R, which satisfy the relations RZ3= R , = 1 ; R’, = R , = 1 ; and R2 R, R, R, = R , = 1. Verify that these elements satisfy the given relations. Let

Show that T(R,) and T(R,) satisfy the same relations as R, and R,. Conclude that there exists a homomorphism T : S , --t GL,(K) determined by this correspondence. Work out the representation T for all elements of S , . Rernurk {*). The representation of a finite group G on its group algebra KG is a special case of a more general construction. Let H be a subgroup of G with index [G : H ] = 17. Let V denote the complex n-dimensional vector space with basis B = {vl = l H , v2 = x2H, . . . , v, = x , H } consisting of the distinct left cosets of H in G. Define a representation T : G -+GL(V) such that T(y) : V -+V is defined on the basis B by T(g)vi= T(g)x,H = gxi H for g E G.

14. Verify that the procedure defined above determines an n-dimensional representation T of G. 15. Find the kernel of the representation T of Problem 14. 16. Let G denote the group of order twelve whose Cayley table is given in Problem 7: (a) Verify that the conjugacy classes of this group are K , = {I}, K2 = {2), K , = (3, 6}, K4 = {4, S}, K5 = 17, 10, 111, and K6 = (8, 9, 12}, (b) Verify that N = (1, 2, 3,4, 5, 6) is a normal subgroup of C. (c) Verify that H = (1, 4, 5 ) and A4 = (1,2) are normal subgroups of G such that G / H is cyclic of order four and G / M is isomorphic to S , . 17. Make use of the information in Problem 16 to determine five nonequivalent irreducible representations of G. 18.

8H

Let H

= (8.

= ( I , 4, 5 ) = v , . 2 H = (2, 3, 6) = v , , 7 H = f7, 10, 11) = v , , and 9, 12) = v 4 . Use the information of Remark (*) to determine a

137

Problems

four-dimensional representation of G on the space V with basis {vl, v 2 , vg , v4).

An idempotent I in the ring K, of all complex n x n matrices is an n x n matrix I such that Z2 = I. 19. Find the set of all idempotents in the ring K2 of all complex 2 x 2 matrices. Use this result to describe the set of all minimal left ideals in K 2 .

20. Use the ideas of Problem 19 to describe the set of all proper left ideals in K,,. 21. Let A denote the algebra of all complex 3 x 3 matrices of the form

1:

a13

ao,, 211.

Prove the set of matrices of the form

constitute the radical R of A.

(1

a13

0 0

a23

0

Show that in the decomposition (3.32), e, is the only idempotent generator of the simple component Ji . 23. Show that if A is an algebra with identity over the complex numbers K then a left (right, two-sided) ideal J is merely a subgroup of the additive group of A which is closed under left (right, two-sided) multiplication. 22.

24. Show that the A.C.C. and D.C.C. holds for ideals of a complex finitedimensional algebra A with identity.

25. Show that any complex finite-dimensional algebra A with identity contains a radical N which is a two-sided ideal containing every nilpotent ideal of A.

Prove that the two-sided ideals Ji contained in the decomposition (3.30) are simple.

26.

27. Complete the details of the scaling of ei, and eni in the determination of the matrix units of the simple components Ji . 28. Show that the trace of a linear transformation T is independent of the choice of the matrix of T.

29. Show that the tr(AB) A and B.

= tr(BA)

for any two complex n x n matrices

2. The Representation Theory of Finite Groups

138

30. The group G of symmetries of the square is the set of all rotations of three-space about the origin which carry the square into itself. These rotations

can be described in ternis of the permutations they effect on the vertices of the square. These are given by

- (;;:)’

’ ) : ; :3! (

1234 1234 - (3412)’ - (2341)’ 1234 (1432)’

1234 (4321)’

1234 (4123)’ 1234 (2143)’

They may also be described by matrices, namely,

The Cayley table of this group is 12345678 21436587 34217865 43128756 56871243 65782134 78563412 87654321

(a) Find the derived group G’ of G and compute the one-dimensional representations of G from those of G/G‘. (b) Determine the complete set of matrices corresponding to the rotations. These matrices present a two-dimensional irreducible representation of C in a natural way.

139

Problems

(c) Write out a full set of irreducible representations of G together with the corresponding character table.

31. Problem 17 can be used to find a complete character table of the group G of Problem 7. (a) Compute the matrices of the permutation presentation of G on the left cosets H , 3H, and 5H where H = {l, 2, 7, S}. (b) Compute the character of this permutation presentation. (c) Reduce this character into its irreducible components. 32. The following Cayley table is that of a group G which is one of the fourteen groups of order sixteen. Its derived group G' i s (1, 2). The Frattini subgroup 4 of a group G is the intersection of all maximal subgroups of G. 2 3 4 5 6 7 8 910111213141516 1 4 3 6 5 8 710 9121114131615 4 1 2 7 8 5 61112 91015161314 3 2 1 8 7 6 5121110 916151413 5 6 7 8 2 1 4 314131615 9101112 6 5 8 7 1 2 3 41314151610 91211 7 8 5 6 4 3 2 1161514131112 910 8 7 6 5 3 4 1 215161314121110 9 910111213141516 1 2 3 4 5 6 7 8 10 9 1 2 1 1 1 4 1 3 1 6 1 5 2 1 4 3 6 5 8 7 1112 9 1 0 1 5 1 6 1 3 1 4 3 4 1 2 7 8 5 6 121110 916151413 4 3 2 1 8 7 6 5 1314151610 91211 6 5 8 7 1 2 3 4 14131615 9101112 5 6 7 8 2 1 4 3 15161314121110 9 8 7 6 5 3 4 1 2 161514131112 910 7 8 5 6 4 3 2 I 1 2 3 4

(a) Find all the one-dimensional representations of G. (b) Use the one-dimensional representations to locate all the maximal subgroups of G. (c) Find the Frattini subgroup 4 of G.

33. Let G be a group of order ten with the following Cayley table. ~

~~~~~~

1 2 3 4 5 6 7 8 910 2 3 4 5 110 6 7 8 9 3 4 5 1 2 910 6 7 8 4 5 1 2 3 8 910 6 7 5 1 2 3 4 7 8 910 6 6 7 8 910 1 2 3 4 5 7 8 910 6 5 1 2 3 4 8 910 6 7 4 5 1 2 3 910 6 7 8 3 4 5 1 2 10 6 7 8 9 2 3 4 5 1

140

2. The Representation Theory of Finite Groups

Show that the derived group H i s (1, 2, 3 , 4 , 5 ) . (b) Use H to find the one-dimensional representations of G. (c) Verify that G has four classes of complex irreducible representations with dimensions 1, 1 , 2, 2. (d) Let T be an irreducible two-dimensional representation of G with representation space V. Suppose that u E V generates a subspace U invariant under TI H where T(2)u = MU. Show that T(5)u = ci4u and that if T(6)u = w, then (u, w} is a basis of V. (e) Show that T(2)u = MU and T(2)w = a4w while T(6)u = w and T(6)w = v. Thus 2 and 6 correspond to the matrices (a)

respectively . (f) Work out the matrices of all group elements. (g) Determine the admissible values of M in order that T be irreducible. Compute the character table of G from Problem 33 and check all orthogonality relations. 34.

35. Let the Cayley table of the group G be given by the following. 1 2 3 4 5 6 7 8 910111213141516 2 3 4 5 6 7 8 110111213141516 9 3 4 5 6 7 8 12111213141516 910 4 5 6 7 8 1 2 31213141516 91011 5 6 7 8 1 2 3 413141516 9101112 6 7 8 1 2 3 4 5141516 9101l1213 7 8 1 2 3 4 5 61516 91011121314 8 1 2 3 4 5 6 716 9101112131415 916151413121110 1 8 7 6 5 4 3 2 10 9 1 6 I 5 1 4 1 3 1 2 1 1 2 1 8 7 6 5 4 3 1110 9 1 6 1 5 1 4 1 3 1 2 3 2 1 8 7 6 5 4 121110 9 1 6 1 5 1 4 1 3 4 3 2 1 8 7 6 5 13121110 9 1 6 1 5 1 4 5 4 3 2 1 8 7 6 1413121110 91615 6 5 4 3 2 1 8 7 151413121110 916 7 6 5 4 3 2 1 8 16151413121110 9 8 7 6 5 4 3 2 1

?(,

The classes of this group are K , = { l } , K2 = ( 2 , S } , K3 = ( 3 , 7}, K4 = (4, 6}, -C5> K --C 19, 1 I . 13, IS}, and K , = (10, 12, 14, 16f. Omitting the trivial I.

141

Problems

case of the identity K , , the structure constants are Cizj

0100000 2010000 0101000 0010200 0001000 o o m 2 0000020

Ci3 j

0010000 0101000 2000200 0101000 0 0 1 m 0000020 0000002

cx4j

0001000 0010200 0101000 2010000 0100000 0000002 0000020

Ci5 j

0000100 oO01000 00 10000 010oooo 1000000 0000010 000000 1

C L ~ J

o000010 000i)002 0000020 0000002

C
0000001 0000020 0000002 0000020 o m 1 o 0000001 4040400 0404000 0404000 4040400

(a) Use the given structure constants to find the determinantal function. (b) Find the character table of G from this result.