Symmetry adapted functions belonging to the crystallographic groups

Symmetry

Adapted Functions Belonging Crystallographic Groups* HAKOLU

V.

to the

MCINTOSH~.

Various authors have discussed methods of obtaining t.he irreducible representations of groups containing nontrivial normal subgroups. in terms of the representations of the normal subgroup and it.s factclr group. These results have applied to special cases, such as to normal subgroups of prime index or to the case in which a subgroup exists isomorphic to the factor grcwp and having one element in each coset of the normal subgroup. These results have been DS 1ended to a general theory, which not only includes all the results of the papers cited, but also enables one to obtain very easily the characters and represrnt:it ions of certain classes of groups heretofore obtainable only by special methods. I~~ram~~l~~s of such groups are the crystallr~gr:~phic “douhfe” grtrups which have been discussed by Bethe, IGliott, and Opecho\?-ski as well as those crystal l~)graphic lattice groups for which there is no subgroup isonlorphic to the corrrspcmding point group. The basis of this theory is a canonical form fof the representations of a semidirect product of two groups; a generalization of the representation theory of direct products which is dwelopcd in the present 1,:lpPr.

When one has a reprrscntation 1‘ of a finite group G acting on a vector space IT, Wigncr (I ) has shown how it. is possible to manufacture projection operators which select t,hc subspaces Vi of I’ which belong to the various irredu&le I-(‘prcscntations ri into which I’ may br dwomposed. AIoreowr, he has shown hog one may selwt a basis in each of t,hese subspaws with respwt to which the t)ransformations of G’ have r’ as t,heir matrix representations. Not only arc tht> projection operators which one may use to sclrct t,hcir basis dewribed, but also the construction of ladder operators by which other basis rlement,s may hc ob tained. onw one of them is known. Nielsen and Bvrryman (sj hare reproduwd his same s, applying to an Vcnkat,arayudu (3) diswssed the of symmrtry functions, while pat,rick (,$I given an whicah may usrd in of cyclic dihedral * Iyniversity. Present IVnivrrsit.,x

at the

on Molecular 1958. Theory Project, Gainesville, Florida.

and

Spectroscopy.

St:itr

Ohio,

address: of Florida.

51

)ep:u?rnents

of Physics

anti Chemistry.

52

MCINTOSH

symmetry. The projection operators were recognized as independent objects by Fokker (5)) who has given as well an intuitive geometric description of a means of obtaining them. His work is interesting in another respect; he has shown how the projection operators may sometimes be factored in terms of simpler projection operators belonging to subgroups. The factorizat)ion of projection operators has many advantages and was first discussed in detail in a paper by Melvin (6). Two advantages make themselves immediately apparent. According to illelvin’s factorization, if a certain function was annihilated by one of a number of factors of the projection operator, then its entire projection vanished and it could be discarded as a source of symmetry coordinates belonging to that particular row of the representation in question, A second advantage is one of conciseness, when for large groups or representations of high dimensionality the labor of listing and manipulating the represent’ation matrices for all the group elements becomes prohibitive. Then it is a great advantage to have a few factors which can he combined in all possible ways to construct a!1 the matrices one expects to use. Twenty operators comprising two sets of ten each, when used to generate a hundred others gives an idea of the saving of labor which could be realized. Not only is there this numerical advantagr, but there is a certain theoretical advantage as well when it comes to classifying the symmetry types, since they can be described in terms of the factors rather than of all the operators. Yet another scheme has been used by Lowdin (7) to obtain the projection operators belonging to the three-dimensional rotation group. In his factorization, the projection operators are written as products of annihilation operators. This factorization stands in contrast to Fokker’s method which utilizes projection operators belonging to subgroups, and &bin’s method, in which one obtains a projection operator belonging to the identity representation of a subgroup, and a second factor possessing no special properties. In certain cases, Melvin’s and Fokker’s results coincide. Melvin’s method was largely empirical, since it depended upon the knowledge of certain special properties of the representation. Since the irreducible representations of a group fail of being unique with respect to an arbitrary similarity transformation, this means that his factorization method does not depend upon the intrinsic structure of the group, as Fokktr’s does. One class of representations to which Melvin’s methods may be applied is composed of the imprimitive representations of a group. Imprimitive representations have been discussed extensively (8)) and are always induced by a subgroup in a certain canonical fashion. The theory of induced representations has been described by Mackey (g), and they account for many of the representations in Melvin’s tables. Imprimitive representations arising from the existence of normal subgroups are especially important. Their significance is simply that the factorization then depends upon the subgroup structure of the group G, and not fortuitously upon properties of its representation.

SYMMETRY

AL)APTEI)

FUNCTIOSS

OF CRYSTAL,L,O(;RAPHIC

GROUPS

21

Many authors hare discussed methods of obtaining the irreducible representations of a group containing a nontrivial normal subgroup N in terms of the irreducible representations of N and its factor group GiN. Boerner (fil) has described the case of a normal subgroup of index 3, whose coset contains an element of order 2, this latter being an unnecessary assumption which he make t,o simplify his proofs. Individual instances of “halving subgroups” have been discussed ; the rotational subgroup of the orthogonal group by Braucr ( 11) , ad the alt,emating subgroup of the symmetric group S,, by l’robenius (I,“). Schul (13) treated the more general case of a normal subgroup of arbitrary prime index, the same case later being discussed independently by Seitz (Id). Seithrr author recfuired that there he an clement. not in N \those order was eclual to t.he intlcx of N. Wintgcn (16) has investigated the case where N is abelian, and his methods have been used by Iioster (16)) subject to t,hc further restriction there be a subgroup H containing just one element from each coset of N. Such a subgroup will, of course, be isomorphic to G/N. &lackey (17) has given a formal mathematical theory of the represcntIations of such groups, which are called semidirect products of N and H. The assumption that N is abelian is unnecessary (18), and t.he requirement that, such a subgroup as H exists can be dropped, but 1vit.h the consecfucnce t’hat now one must, know t#he projective representations of rV and G/N (19). In complet,e generality, it is possible to obtain the irreducible representations of any finite group with a nontrivial normal subgroup in terms of irrcduciblc reprcsent,ations of its normal subgroup and of irrcduciblc project,ive rcprcscnt:lt,ions of certain subgroups of the factor group. These representations arc ob t,ainrd in a form to which methods analogous to Melvin’s may bc used to obtain the projection operators in a factored form, However, the fact,orization most nearly resembles Fokker’s results, since t,he factors are combinat,ionx of projection operators or ladder operators belonging to the normal subgroup or to the pro jcctivc rcprescntations of subgroups of the factor group. The groups which can be so treated comprise a number of interest in physics. Among these are the crystallographic “double” groups of Bethr (20), Elliott (%‘I), and Opechowski (92). They possess normal subgroups of order 2 rather than index 2; their factor groups are the ordinary point groups. Also included are t)hc cryst,allographic lattice groups, especially those for lattices which contain glide planes or screw axes for which there is no subgroup isomorphic to the point group, and in fact our results supplement the classical paper of Bouckaert, Smoluchowski, and Wigner (23) by making this &ension of their results. More recently, this theory has been summarized in the textbook of I,omont (Li), which sketches the theorems without giving the derivations or proofs. Also, Zak (25) has shown how one may obtain the characters of the nonsymmomorphic space groups directly. The organization of t’his paper is as follows. Virst, wc define the semidirect

MCINTOSH

03

product of two groups, and discuss some properties of a semidirect product. We then treat some general properties of sets of conjugate representations, in preparation for the demonstration that the irreducible representations of a semidirect product may be derived from those of its factors. A canonical form for the irreducible representations of a semidirect product is thereby obtained. The applications of this theory to constructing factored projection operators have been discussed in a previously published paper (26), and include the derivation of the characters, projection operators and transition operators of a semidirect product from those of the factors of the group, as well as a similar treatment of the problems of reducing direct and inverse Kronecker products, determination of the equivalence of a representation to its complex conjugate, and the construction of symmetric Hamiltonians, by referring these to corresponding problems concerning the subgroups. Unfortunately, the work of the present paper is incomplete, due to the possible occurrence of projective representations belonging to the little groups, and the theory derived is valid only insofar as these representations do not occur. Lomont (24) avoids this difficulty by considering little groups of the “second kind,” but the fundamental difliculty remains, since knowing the ordinary representations of his little groups is equivalent to knowing the projective representations of ours. An example of the breakdown of our theory, as a general means of construetiny (rather than deswibing, the task to which Mackey (19) addresses himself) irreducible representations of semidirect products is afforded by the semidirect product Q: I74 , where Q is the quaternion group, and V, is the four group, whose generators act on Q by changing either i or j into its negative. The task of surveying the theory of projective representations is reserved for another paper. By developing the theory of semidirect products in terms of the convolution algebra of a group, the present paper yields the proofs of the theorems assumed in our earlier paper on factorized projection operators (26)) and in a form amenable to subsequrnt usage by a symbol-manipulation program designed for a high speed electronic computer. THE

SEMIDIRECT

PRODUCT

Let 2’ be a group and let P be an operator group for 7’. Then, by the semidirect product of T and P one means the group T: P composed of elements from the set T X P for which multiplication is defined by the rule:

(t’, P’)(4 P) = G’.P’(t) >P’d .

(1)

In this definition it is worth emphasizing the fact that the pair of group elements t and p have been written in the opposite order to that customary in many physical papers. l;or purposes of application, one may imagine that T is the subgroup of translations of a crystallographic lattice group, while P is the point group. Then, in the pa,ip ct. p) the translation has been written first, followed by

the rotation. This notation has been adopted primarily to avoid making of detinit~ions and derivations unntwssarily clumxy. To wrify that T: T’ is actually a group WC not’c that (t”, $9;

(L’, p’,(t,

P)l = (t”p’w’p’(f)

a numbw

1, l”$f})

n-hi 1~’ I T as well as the associativity of multiplication in bot)h P and T, IVCsee t-hat’ t)hc associative law is verified for thtl semidirwt, product. The identity in T:l’ is (c, V) sinw (f,,P)(f,pJ

= (c.c(t),e.p’)

= (t,p).

Th(s c~cfunt~ion: tip-‘m-‘, shows t,hat it, p)-’ ficd. Sinw

p-p tt, p) = (ip-‘(t)j=

(!~~‘(t)}--‘,

‘p-‘(t),

p -‘IN = ((3,e)

p -‘). Thus all the group postulates

are satis-

\vc SPPt,h:lt I’* = ; (t, p) E T: I’ ! t = r/ is :L subgroup

of T: I’ isomorphic

to I’. Similarly,

sinw

(f’, P) (f, 0) = if’? (f 1~ cc,) = w

SW

(f’f, cl,

that T* = j (t, p) c, ?‘:I’ / p = cl

is a

subgroup which is isomorphic* t,o T. ‘I’hc cwniugatc of an clemrnt Ct, p) is givrn (11,S-‘(t,

by t’he formula

p) (z., s) = (Sc(L’-‘tp(l’)

), s--‘ps).

(2 1

From this it follows that T* (but not necessarily I’*) is a normal subgroup of T: P. The fac+,or group T: Pi T* is isomorphic to Y by thtx mapping p + (e, p) T*. WC havct T: I-’ = T*P* in view of the facttorizat,ion

(c, p) (p-’ (f),

ei = tt, PI,

56

MCINTOSH

so that the alternative factorization !l’: 1’ = P*V* is valid as well.’ It is obviously true that P*fI T* = i (e, t)), and finally we note that (e, p) (6 e) (e, p)- ’ = (p(t) , e). The meaning of this last remark is that not only is P an operator group for T, but that I’* is an operator group for T* as well, operating through inner automorphism in 7’: P. With these preliminaries we turn to the converse problem of determining the circumstances under which a group G is isomorphic to a semidirect product of two of its subgroups. These requirements are three in number: 1. H is a subgroup, N is a normal subgroup ; 2. NH = G; 3. N fl H = {e). We may regard H as an operator group for N by conjugation; that is h.(n) = hlbh-’

h E H, n E N.

It is necessary for N to be normal if we are to be assured that h (n) is an element of N. If we now make the mapping from N: H to G,

and note that (n, h) (4, h’) = (nhn’h-‘,

hh’),

we find that (72,h) (n’, h’) --f nhn’h’. Thus it is verified that the mapping is a homomorphism. To see that it is an isomorphism, we show that every element can be uniquely factorized as g = nh, g E b, n t N, h t H. Suppose on the contrary that t,hrrc were two factorizations: g = nh = n’h’. Then h’h-’ = n’-ln * Since P*l’* = Y*P* = 7’: P, and their intersection is the origin, we see that 7’: P is the “weak direct product” of P*T*, as it is sometimes called (see footnote 9 of Ref. 6). However, the “weak direct product” and the “semidirect product” are yuite different concepts, the weak direct product being more general, since it does not require one of the subgroups to be normal. A counterexample which distinguishes these two products is afforded by the tetrahedral group. Choose one subgroup of order six comprising the symmetries leaving one point fixed; the other as the set of powers of one of the elements of order four. The tetrahedral group is their weak direct product. Neither is normal.

and an clement, of H would be equal to an element

of N. This

only happen

cm

when both are the identity, and thus only when h’ = II, 11’ = tb. In part’iculzu the factorization C’ = P .e is unique, and only (P, c) can map into the idcntit,y. Since

t,hc kernel

of the mapping

contains

only t#he identity,

the mapping

is an

isomorphism.

It is cust.omsry

to indulge

a slight

lack of prwision,

a srmidirwt product of two of its subgroups, is isotrtorphic IO their semidirect product. The three subgroups that

post’ulates

by which a group is a semidirwt

may bt summarized

in the following

the right, c*osets of N generated

same is true of the left (wets elcmcnt

and say that

of

H

fashion.

by clcments

gcnerat,ed

of

!/,

product

of two of its = G, IVP SW

fill up the group G. The

H

by N. Since N n

of II wn lie in each right coset of N, and c*onwrsrly, thll

brloug to N/r II

.I’!/

I

of H. The reason is simple-if N flH, the cluotient belonging to H H and hence to H; and to N by the definition

.1.!f

’

E

= (1;.r = y and the contradiction

L%‘it,h t,his results. our summary subgroup

of (;;

(aross swtion

H

tnkcs

the form of saying of its co&s.

wit,h respect, to N is to mean that

H

H = ie], only

,(I, 11 E NU fl H, bwause t,he fac~t,ors

that, N is a normal

To say that

contains

is a right

H

precisely

one clcmcnt

from each right, coset of N. The ~~onscquencc of this case, as distinguished the (‘;tsc whew no cross section subgroup csisL

isomorphic? to t.he fwtor

Although left

(TOSS

coincide.

is a subgroup,

as n-~11, sinw

group G/N. Swh

a subgroup

by t,hc swond

postulate

upon the apparent

of the criterion

time w* shall SW the reason

H. ,Ic N, w('may

of a normal

subgroup

product.

implied

_At t,hc same:

(II, h) in that, order to define :t

.wmidirwt prodwt,, rather than in the order (II, ‘tl J First, although we write NH = 0, WC caould instead iug /j C

need not always

lark of symmetry

for n semidirect

for t.aking the pair

has a

of the cosets of N, itf is a

the left, and right, oosets

Thus n-r should comment

from

is that, G then necessarily

WC have said that II is n right cross swtion

section

OIIV

only one element

of right coset. Thus our assertion.

cstablishw

is a right’ cross swtion

t,hat it

Since NH

of N CL~ lict in each left, cosct .r #

a group is

when in rralit,y it is meant

write

HN = G. I:or,

tak-

writ{>

trh = h (Vtrh). Sinw N is normal, h-‘&l it N, and thus whenever we have NH = write HN = G,and comwsrly. The reason for stat,ing the postulate

G, we mu)

with N in the preferred position on the right, is to make the mapping take the ximplcr form (71, h) + rzh, rather than (tL,h) - h.h-‘01). Underlying this prefcrcnce is the decision to write the semidirect

N:

as we did,

H - G

prodwt

in

t.hc form N: H with pairs of elements (tz, h) rather t.han in the form customarily used by physicists, which is (h, tt,). Tho reason is the following. The prototype of a semidirect product is the I~;wlidean group, comprised of t,he rigid mot,ions ot an Euclidean space--say our ordinary thw-dimensional space. If .r is a \-wtor,

MCINTOSH

58

an operation of this group is defined (see Ref. 16, p. I75j 2’ = Rx + t,

by

(3)

where R is an orthogonal matrix and t is a fixed vector. Icrom this definition it is clear that the rotation is performed first, t,hcn the translation. Otherwise we would write I P = R(s + t). When two such transformations such as the one in Ey. (3) are compounded, the resulting expression shows that the Euclidean group E3 is the semidirect product of the three-dimensional vector space V” and the t,hrer-dimensional orthogonal group O3 . Thus E3 = V3: 0, . Ihe to the fact that we usually deal with groups of operators rather than with groups of operands, me are in the habit of saying that the rightmost operation of a product is to be performed first; and so on, reading from right to left. Since, in t.he Euclidean group we perform the rotation first, then the translation, they should be written in the order (t, R) and we should use the mapping (t, R) + t. R ( ( .) denoting the Euclidean group multiplication, not matrix muItiplication) in order to preserve this order. The resulting simplification of formulas and derivat,ions is our reason for violating the usual physicist’s convention. CONJUGATE

REPRESENTATIONS

Let us suppose that r = (D (t, p) ) is a representation of the semidirect product G = l’:P. The matrices of r may be factored in a manner corresponding to the factorization of G into T*P” = G: D (t, P) = D (t, p)D Cc, P). Let us adopt the distinctive

(J)

notation D(t,e)

= T(t),

D(e,p)

= P(P).

(5) The two sets of matrices 0 = ( T(t)) and II = {P(p)) form representations of T and P, respectively. Since P* is an operator group for T* by inner automor. phism, the matrices of T must satisfy the requirement P(P) N)PW’)

= T@(t)).

(6)

Conversely, if 6) and II are two representations of T and P, respectively, whose matrices are conformable and satisfy this relation, then it may be verified that D&p)

= T(t)P(p)

forms a representation of G. This condition is then a necessary and sufficient condition for a representation of G.

SYMMETRY

Equation

ADAPTED

(6)

FUNCTIONS

be rearranged

T (p(t))

W(~))~(p(t’~)

GROUPS

.ig

read = T(p(t))Pip).

P(p)T(f) Now, the matrices

OF CRYSTALLOGRAPHIC

satisfy

(7)

the equation = Vp(t)p(t’))

= T(p(tt’)),

which shows that they also form a representation of l’. This new representat,ion may be either equivalent or inequivalent to 0. In either event, however, it, is clear t,hat it is reducible when 8 is reducible, a.nd irreducible when 0 is irreducible, since the two representations have the same matrices, differing on1.v in t,he assignment of their arguments. ltat,hcr than assuming that p operates on the argument, we may assume that p operates on T(t) itself to give a new representation with the same old argw merits. Thlls, n-e define’ p-‘(C)) t,o be the representation p-‘(G))

= ; T(p(t)

)}

whose mat.riccs we designate by p-l (T(t) ) . T ~$0 such as (4 and . reprwwtationa, p-’ (C-1)) which arc related to one another by having mutually conjugated argumcnts, arc said to he conjugate to one another. The set of all representations conjugate to a given representation 0 is called the star’ of (3. The representations of a star may or may not be mutually equivalent. In any event, they may bfx assorted int,o ryuivalenct classes of mutually ecluivalent representations. The cxclui\,alencr classes are called the prorbgs of the star. It should be noted t,hut thr star of a representation is not an inherent propetty of tht represent,ation itself, but rather it also depends upon t,hc group of automorphisms which is being uwd t,o form the semidirect product. The set of all p E P for which ~~‘(8) is cquivalcnt, to ct) is called t,he little group of (4. It is designated by enclosing 0 in a pair of brackct,s: [@I. Having assumed that (4 and p-’ ((3) are equivalent, we may write

p-V’(f)

1 = T(p(t))

= p(p)T(f)p(p)-‘.

2 Professor Melvin has kindly pointed out that the mathematically reasonable definition p-l(&)) = /7’(p(/))} rather than ;u(@) = 17’(p(t))l is a sllbtle point worthy of clarification. The definition is arbitrary, but motivated by a desire to have P operate on (3 homomorphitally rather tharlantiho,)lon~o,ph~~f~~~~ [j(ah) = f@),f’(n)j. We have Il’(pq(t)) = (pp)j7’(t)) on on the other, if we adopt the latter definition It is the one hand and y’(p.q(t)) = y(p(T(t)) a qaestion of w-het,her p transforms the futwfiorz 2’ or the argument t, and it is desirable to have t,he definition so phrased that it may be regarded interchangeably in either light. 3Macltry (17) uses the t,erm orbit; we have chosen the more common physical t,ernli1mlogy”3 although the direct, physical significance no longer persists when (-1 is not on+ climensional. These representations, if the lattice groups are the one-dimensional irreducible representations of the translation subgroup, and are characterized by certain wave vectors in the reciprocal lattice. All the wave vectors belonging to a star could be gotten from one of them hy carrying out the symmrt,ry operations of the lattice, resulting in t,hr rn:tn>‘~ lwongetl fiyrlw called the “star.”

60

MCINTOSH

When 0 is irreducible, then by Schur’s lemma, the set of matrices 44 = (cl(P)] is determined except for a nonzero scalar multiplier, which may be different for each matrix p(p) . Since we deal with finite groups whose representations are finite dimensional, we may use the auxiliary condition that each matrix p(p) have unit determinant to further determine the set M. In that case it forms a projective representation of [O]. The little groups of all the prongs of a star are conjugate to one another. For, if we have 4-* (0)

= cr(Y) @P w’,

then (q-l)

= r (/l (y) o/J (y)_‘) .

- 5 (0)

now, 44n)U~)P(p)-1)

= /4Ym-1e4Y)-1

=

P(YhvwMY)-l,

and thus we have (qr-l)-lY(o)

= (y)?(O)&)-‘.

Since the converse is also true, it is thereby proven that ?qqr-’

= [r(6))]

(9)

or, as we might say:

f-WI) = b”(@)l. Having selected one representation of T,such as 9, as a reference representation, we find that its prong is generated by its little group, since it contains the representations .$-‘(0) for E E [O]. The other prongs of the star of 0 are generated by the co&s of the little group. Letting p be the generator of a coset, we find T(E@(EC’)

= P(E)T(~S~.~~-‘)M((F)-~ = A~)P(N))P(E)-‘.

That is to say, the same set M defines the mutual equivalences of all the representations in each prong. The various properties of the star of a representation are summarized in Fig. 1. The corresponding little group is shown in Fig. 2. REPRESENTATION

OF THE

SEMIDIRECT

PRODUCT

We have seen that the matrices of a representation r of a semidirect product G = T:P satisfy Ey. (7)) P(P)TO)

= I’m’,

(7)

SYMMETRY

ADAPTED

FUNCTIONS

OF CRYSTALLOGRAPHIC

GROUPS

61

Frc;. 1. The star of in representation

P\* I

r-+LLL-LL4

,

,

I

,

RIGHT COSETS OF C81 FIG. 2. A semidirect product T:P showing the normal subgroup he subgroup P* isomorphic to P.

Z’* isomorphic

to 7’ and

62

MCINTOSH

where P(p) and T(t) are the matrices defined by Eq. (5). It is convenient to assume t,hat we have chosen, among all the representations equivalent to I’, such a one that 0 at least is completely reduced, and such that further, if several equivalent irreducible representations of T occur in 0, they are actually equal rather than merely equivalent, and that they occupy contiguous positions along the diagonal when 0 is displayed in its completely reduced form. Then the matrices of II may be partitioned into submatrices to correspond to this reduct.ion, and Eq. (7) may be examined with respect to this representation. Three levels of submatrices distinguish themselves during the preparation of I? in the canonical form which we have just described. In the matrices of 0 one considers first of all individual matrix elements; then the submatrices which form irreducible representations Oi of T, and finally the groups of these submatrices which are equal, due to the fact that Oi may occur with a certain multiplicity. A similar partitioning is induced in the matrices P(p) . Thus, it is convenient to give the basis vectors of the carrier space of r three indices. The first designates a subspace stable under a certain irreducible representation Qi of T. Since Oi may occur with the multiplicity ?)I, the second subscript, running from 1 to ~1, serves to distinguish the minimal stable subspaces belonging to the same representation from one another. The final, third index, distinguishes the basis vectors within a minimal stable subspace. Written out matrixwise with respect to such a basis, Eq. (7) takes the form shown in Fig. 3. The submatrices of T (tj all vanish save those on the diagonal, and they are given two indices, Tj”(t) indicating the jth occurrence of ith irreducible representation, Oi. The submatrices of P(p) have four subscripts, Pzj,i’j, (p) indicating the submatrix connecting the jth occurrence of the ith irreducible representation Oi’ to the jth orcurrence of the ith irreducible representation.

FIG. 3. In terms

of these submatrices,

Eq.

Equation (10) (10) may be writ,ten out elementwise to

obtain P
(p).

(11)

Thus for a fixed p, Pii,,‘j, (p) satisfies the hypothesis of Schur’s lemma, by intertwining two irreducible representations of T. As a consequence, either

SYMMETRY

ADAPTEl)

FUNCTIONS

OF CRY8’~41,I,O(:RAPHTC

CiROUPS

(i:<

f ij,,,J, (p) = 0 if 0” is inequivalent to p-‘(8’) ; or if these two representations are equivalent, Pi,,i, j* (p) is nonsingular if it is nonzero. If WC should presume that I? is irreducible, and we sele(at a certain pair of the irreducible representations of T, say 0’ and O”, then WC cannot allow (4” to be inequiralent to p-‘(0’) for all p. Otherwise the matrices appearing in &:(I. (10) which generate I’, will generate r in an explicitly reduced form. When the irreducible representations (4’ into which (3 decomposes are assorted into stars, this result means that for r to be irreducible, only those representations may occur which belong to a single star. On the ot,her hand, E(l. (6) shows us that (4 and p-‘(k)) are equivalent. Whether we write I’(t) or T (p (t) ) , we obtain the same matrices with permuted arguments. Thus when 0 is completely reduced as me have assumed, p--‘(H) is completely reduced as well. Xow, the reduction of a representation (or it.s qui\-aIcnt#s) into its irreducible constituents is unique, except’ for the order in which the irreduc4ble components appear and their possible replacement by equivalent, representations. Thus, when we select the irreducible component p-’ (0’) from pmI ((4) , w know that it must’ be equivalent to some @‘(taken from (4). -1s this result. is t.ruc whatever p is chosen, we see that representations from every prong of t’hc star of 0’ must occur in (9. Moreover, if p-’ (@Ii) o(*curs in p-’ ((9) with a certain multiplicity, then (9”’ occurs in $ with the same multiplkity, since t,hct number of times a certain irreducible rcpresentat,ion occurs in p-’ ((9) is not (*hanged upon going to the rcluivalent representation (5). SintBe this is just, th(, same us thr number of times that @)’O(YUI’Yin (4, n-e see that, all the prongs of :I certain star orcur in CA)whenever one of them does, and with thn same multiplicity. When I’ is irreducible, the prongs of rxactly one star o(acur. We may summarize these results by displaying an explicait form \vhicsh the> matricbes ?‘(f 1 must possess. Suppose that a complete and nonredundant set ot rightkoset generators has been chosen for the lit’tle group of one of the irreducible representations-say (3”. Ilet these generators be c, , c2 , . . . , ck , with c., = C. This amounts to a selection of one standard representation from each prong ot the st’ar, and these shall be the representations (4’, k?, , G”. The digits 1, 2, . , 1~shall be the first index dcscrihing t,he basis of t,he carrier space 1’ of I’. The second index will have the same range, say 1, 2, . , P, whatever the \Aue of the second index. P is the uniform multiplkity with which eac+h reprcsentat,ion Wi occurs in t,he reduction of (3. The third index shall hn1.e the range 1, 2, . . 1 , 111, where ~1 is t,hr uniform dimension of the irreducible representation Hi. Then 7’ (t ) has the caanonical form shown in Fig. 4. When t,he matrix T’(t) of the irreducGble rcpresrntation 0’ is repeated several times in the intermediate submatlix, WI may write it as a tensor product T’ (t) 0 1 with the unit matrix, whost> o&r is the multiplicity f. I’(p) will at the same time have a submatrix structure whose form Iv-ill he det~erminc~d b,v Qchur’s lemma from that of T(f). As we hare seen, Pij,i’>, (p) = o if (4” is inequivalent t’o p-’ (Oi) or, if these represent,atlions are equivalent,, t,hcn

64

MCINTOSH

T

T’(t)

ml

i T(t)

=

I”

T3(t)

Ik

--uI

-_-

i

I_

FIG. 4. Canonical

1

form of 7’(t)

P
= p-‘c;‘c;,

(0”‘)

(where we understand that 0’ = ci’(OJ) , OJ being the reference representation), then for p-‘(Oi) to be equivalent to Oi’, we must have P’Ci’ci~ E [O”‘!. By writing this criterion in terms of the reference representation @‘, we find that it may be written cipe?

E [O”].

(12)

Presuming this condition to, be fulfilled, let us write Cipcy’ = .$; then we have p-l (Oi) = cL!;‘f’ci (Oi> = CT?(p (E) @p(t)-‘) = blew’&-‘. This means that P
P;j,i’j,(p)

=

0 ,r \~ij.iPj,(P)P(tti’)

-

cipc$

E

[OJ] (13)

C*PCil

=

Eis’

9 4air

E

l@“l.

The subscripts for the scalar multiplier X indicate that it may vary between different pairs of representations equivalent under the influence of p, while the subscripts for t indicate that the solution of the equation cipc7’ E [@I may vary

SYMMETRY

ADAPTED

FUNCTIONS

OF CRYSTALLOCiRAPHIC

GROUPS

6.5

between different irreducible representations although not, for the diRerent OP currenees of the same one. Equation (13) shows that P(p) has the form of an imprimitive matrix (8). If we designate by Pii, (p) the intermediate submatrices which correspond to t’he decomposition of the carrier space I/ into spaces stable under the irreducible representations @ji of T, without the distinction bct,ween multiple occurrences of the samr representation, then we have Pi,, = 0 unless p-’ (0’) is equivalent to 0”. Only one such submatrix in each row and column of P(p) is nonzero, and all the elements y C P having the same nonzero submatrix in a certain row 01 column are those belonging to t’hc same right cosct IO”]p of the Mtle group of (4’. I-sing the formula /‘ii’ (pq) =

c

Z’,k(P)Pi.c ((I)

for the mat,rix elements of a product of two supermatrices, and realizing that t.he summands are zero unless simultaneously

t,hat is, unless (~y)~‘(@~)

G O”, we see that WC (aan simply writ,e PZij (pq) =

Pik(pjPkis (q).

ilr,,

It is to be noted, in applying this equation, that k cannot be chosen arbitrarily, but rather it. is the unique index satisfying Eq. (11). When p and q both belong to the little group of G’, WCmay write I’Li(YJPii(rl) = I’>((Prl), whence WC SW that the set of submatrices {I-‘,, (pi 1p E [O”]f forms a rcpresclntat,ion of [(C)‘J_In particular: Pii

= I,

a result we might also have known hccausc I’,, (P) is a fragment, of the diagonal of P(c). Assuming Pii, (s) # 0, I’,,(

(sjP’,fi(.3-‘) = Pii(e) = 1.

Sinc*r all t’hc submatrices of I’(s)

are square, this means that

Pii, (9) -I = P,r j (SC) . I,ct IIS finally turn out attention to the factorization:

Pi? (PI = Pi.,(C,‘)P,,(C$C?) =

I’.,!, cc,4

I’.,,(c,)~‘P,,,(c,pcrl)I’.,,, (C,‘).

(I(i)

MCINTOSH

66

Both of the factors at the extreme ends of this expression may be removed by a similarity transformation; namely, by the supermatrix

Pjl(Cl) 1L =

PjZ(CZ)

. . . .

To determine the effect of this similarity transformation upon 0, and hence upon I?, let us calculate the submatrices PJi (ci) . By Eq. (13)) we find PJj,iP (Ci) =

X.lj,ijs(Ci)P(e).

Calling XJj,ij, the matrix element of a new matrix Ar; , we see that P JJ,.IJ’ (Ci) = 1 0 ilJi(Ci) and thus that the similarity transformation of 0 by IT will leave 0 unaltered. Thus, without loss in generality we may take PiJ(ci) = 1. The significance of this assumption is most easily seen by considering the matrix elements Pji (p) , where p E [Oil. Since all the little groups are conjugate to one another, and since our choice of PiJ (c,) ensures that Pii (p) = PJ, (c; pci’) , we see that we are using the same representation both for [Oil and its conjugates. In other words, instead of selecting arbitrary bases for each subspace of the carrier space V belonging to different prongs of the star of OJ as we are allowed to do because P,,(G) is an arbitrary nonsingular matrix, we have chosen to select the same basis in each of these subspaces. Once this decision is made, we see that the representation n comprising the matrices P(p), is an induced representation (9), induced by the representation {P,,(p) ( p E [@“]} of the little group of OX. The matrices of this representation are also described by Eq. (13). We have PJi..Jj’ (P) = Xji’ (P) I-r(P) * p (p) is the unimodular representation of [OJ] determined by Schur’s lemma from Eq. (11)) and the scalar multipliers Xjjf (p) describe the ambiguity in the solution of this equation. If these quantities are regarded as matrix elements of the matrix R(p) , we see that (p) is a tensor product PJj,Jj,

PJi,.JY (P) = CL(P) 0 A(P)

.

(17)

At this stage, since w(p) is a member of a projective representation, we can only say that h(p) must belong to another projective representation, whose multiplier is the inverse of that of the first representation. Although we can sharpen our results somewhat, by distinguishing a subgroup of the little group for which the conjugate representations are actually equal and not merely equivalent, a study of the example Q: Vq mentioned in the introduction shows that we cannot invariably obtain sharper results, and to proceed further in this

SYMMETRY

ADAPTED

FCNCTIONS

OF CRYSTrlLLOC;RAPHIC

C:ROUPS

67

direction WC should have to study the theory of projective representations in more detail. Although this hiatus does not affect the remainder of this sec%ion, the nest section remains incomplete to the extent that without a theory of projcetive representations, we have no orthogonality rules to guide us in proving that all representations constructed by the methods uncovered in this section are irreducible, and thus generally in establishing the converse of the present section. It’ A = I d(p)} is not irreducible, then once again we may exhibit, I‘ in an explicitly reduced form. By the use of Elys. (16) and (17)) II may be seen to be reduced. while due to the form of t,hr t,ensor product, the form of (3 will not counteract, thr reduction of II. If I’ is tohe assumed irreducible, t,hen A (p) must by an irreducible representation Ak of the littler group of eJ. This diwussion may be summarized by displaying the canonical form for t,hc matriws I’(p). Their matrix elements are l’ijk,,,j,k, (p) . All matris elements vanish unless p-’ (0’) z (4” , so that we have the preliminary redwtion into submat,riws c-onnccting the prongs of (3“ cquivalcnt under conjugation by p. These submnt~riws are themselves tensor prodwts, t#he expression of whirh utilizes the remaining two index pairs, (j, j? belonging to the irreducible reprcsentation A”, and whose matrix clcments arc characterized by the fact that they connwt the jth occurrence of the represent,ation 6;)’ with its j’t,h ocrurrcnw. F’inully, the indes pairs (X, k’) belong to t,he matrix elements of the reprcsent,ation Al, which reflects the fact t,hat p-’ (@‘) is not @I”, but merely equivalent, to it,, and t.hat t.his transformation must. he made before t,hcy can be c*ompawd. Since all matrix elements are referred to the single refcrrnce-representation W’, the arguments appearing in the submatrices are not p, but rather CJIC~‘, when the suhmatris c~onnect,s prong i’ to prong i. The canonical form of I’ is shown in liig. 5. Not only is II an induced represcnt’ation of I’, hut r itself is an induced reprc‘stwtation of G, induced by the representation

’ Strictly speaking we should write 7’*[(+‘]. However, we are neglecting t)etween G and 7’: P, that t.hey are isomorphic ancl not eclud.

the distinctiorl

MCINTOSH

P(P)

=

FIG. 5. Canonical

form of P(p)

while

AJKOP$1, PP’)= 1TJOP$1)P (PP’) I 0 1AK(PP’> 1. The crucial step in this proof is the remark that p (p) TJ (t’)p (p)-’ = TJ (p (t’) ) , which follows from the fact that p(p) represents the little group of OJ. By multiplying out the matrices in the equation D&P)

= T(~)P(P),

which are now assumed to have their canonical forms, we see that D (t, p) has the submatrices 0 A”(c,pc,?)

{ !?J(c,(l))p(c,pc,~)) 0

c,pd

t

[@I

otherwise

(19)

SYMMETRY ADAPTED The nonzero

submatrices

FUSCTIONS may be written

A”“@,(1),

C,PC,$

OF CRY8TALLOGRAPHIC

C;ROUPS

6:)

in the form

= AJK( (e, cm) (t, P) k, cm,)-‘)

and are nonzero when (e, c,) (f, p) (e, c,r)-’ c T[(9J]. Since the cosets of 5”[W’] in G are generated by the elements (e, cm), while the element)s cm generate the cosets of [W’] in P, we SECthat. r has precisely the form of an induced reprcsent,ation. Ss a consequence of this analysis we have seen how every irreducible reprosentation of a semidircct product is an induced representation, induced in a certain canoniral fashion from an irreducible representation of the corresponding little group. It now remains to investigate the converse question of whether or not this analysis has been sufficiently detailed, so that one may say that every representation of G manufactured in this fashion will be one of the irreducible representations. It is to the partial affirmative answer t,o this question that t,he nclst, section is devoted, assuming t,he absenc.c? of projectivr representations. THE SYNTHESIS

OF 1RREI)UCIBLE REPRE8ENTATIONX PRODUCT

OF A YEMII~IRE(I’I-

Ry the discussion just completed, we have seen that every irrcdurible repro’sentation of a semidirect product, ?“:I’ must assume a certain canonical form; to wit, one must take all the irreducible representations of T and assort them into equivalence classes of equivalent representations under conjugation by the elements of P. The set of representations conjugate to a given representation 63 composed of the matrices 7’J (t) , is called the star of W’. Its prongs arc composed of mutually equivalent irreducible representations, while the representations of diffcrrnt, prongs, although conjugates, are inequivalent. Once a certain star had been selectled, and one of its representations chosen, one had t,o find the set of all tht> transformations in I’, under conjugat,ion by which 0” remained e(iuivalent. t#o itself. This was the little group [W’] of W’. Among t.he irreducible projective representations of the little group those have to be chosen whose multipliers are inverse t,o that. of the representation iI/, and among these one had to be chosen, from which a representation of the subgroup T[W’] could he manufactured as a tensor produc&t.. If one had found t,hat TJ (p(f) ) = pJ (p) TJ (t) pJ (p) -I, whercb .1/ = { pJ(p) ] was a unimodular representation of [&I, and that A” was t,hf* irrcducihle representation of [@“I needed, t,hcn A”li (1, p) = T“(t) F” (p) @ AK(p) was a representation of T[W’] mhirh would induce the irreduriblr representation I’ of G. It, is our object now to show that any representation whatsoever of G manufactured in this fashion from a star of T and an irreducible representation of its little group will be an irreduc4hlr representation of G, and that furthermore

70

MCINTOSH

representations arising from different stars, or from different irreducible representations of the little group are inequivalent. Once this is done, the irreducible representations of a semidirect product will be completely characterized in terms of representations of T and of subgroups of P. The present proof is incomplete, insofar as we are considering only those representations manufactured from nonprojective representations of the little group. As a preliminary step we show that irreducible representations of T and of the little group yield inequivalent representations of T[OJ]. The character of AJK (t, p), xJK (t, p) may be found from Eq. (20) ; it is x.‘li (4 P) = 2

G 0) da(P) XK(P),

where x”(p) is the character of A”, and the other matrix elements have their obvious significance. Applying now the formula by which we test the irreducibility of a representation, we calculate

$T

XJK(t,p)x”“(t, p)-’ = (AJK1AJK) =P

1 c OTO[OJ] t,p

t:dt)&(p)xK(p)

u,b.a’.b’

t~‘b’(p-l(t-‘))~~f,‘(p-l)XK(p-l)

1 =

___

c

“TO[OJ] t.p

t~~~t)d~~~-‘)XK~P>X”~~-l)~~~

a,b,a’,b’ cd

Carrying out first the sum over T, and using the orthogonality relations for matrix elements belonging to different irreducible representations, one finds first of all that this expression reduces to:

Next, performing the summation over a, b, CL’,b’, c, and d, we recognize that all the matrix elements of the p’s combine to yield the matrix element of a certain matrix which turns out to be the identity, and when another summation takes the dimension of OJ. The final result is a sumits trace, the result cancels mation : tJ

JK (’

JK 1 a

,

6JJ,

This final expression proves our assertion.

x^(p)x”(p?)

=

1.

(21)

SYMMETRY

ADAPTED

FUNCTIONS

OF CRYSTALLOGRAPHIC

GROUPS

‘71

sow that we have verified the fact that the AJK’s constitute irreducible reprosentations of T[@], we proceed to the question of whether these representations induce inequivalent irreducible representations of G. The irreducibility of an induced representation depends (9) upon first, that the base representation is irreducible, and secondly, upon whether the two conjugate representations (4 PI --f A (4 P), (t, p) ---f A (s-j (v-‘tp (u) ) , s-‘ps) of (21,s)-‘TIOJ](u, S) fl T[@lJ] possess common irreducible components or not,, for (11,s) 4 ?‘[@“I. Since s is an automorphism of T, and v, t, and S(U) all belong to T, \ve SW tjhat (a, s)-‘T[@‘J(v,

s) i-l T[O”] = T([&]

t-l [s-‘(O’)].

The significance of this result is that the intersection is still the product of 1’ by another subgroup, so that when the characters of the two representations, A and its conjugate, are compared over this subgroup, the formula expressing the numhcr of their common constituents will still involve a sum over V. In fart this formula is (letting I; stand for the intersection [O”] fl [s-l (0”) j) :

of &

(~‘tp & (s-l

Substituting oht,ain sum

is expanded as (U,) 1 = c I

c ti, u

element’ of t:, (SC’0) t;b (SC’(P and carrying out the sum

we find ) 2’ first,, IVP

and in out this sum must recall that not belong to little group of whereas to and to little group of s(OJj as well. Therefort> the two factors of each summand of this expression are the matIris elements of t,wo inequivalent irreducible representations of T, and the sum vanishes. With it, the sum (22) also vanishes, and it is proven that every representation of the semidirect product T:P manufactured according to our prescxription is an irreducible representation. Our second and final task is to determine when two irreducible representations manufact,ured according to the prescription will be equivalent. Once again the clucstion reduces to the consideration of conjugate representations with respecat t,o a subgroup. Suppose that we have two representations AJK and AJ’“’ which

MCINTOSH

72

are tensor-product representations such as we have described, representing the subgroups T[@] and TIOJ’], respectively. Then the criterion for whether the representations of G which they induce are equivalent or not is whether or not there is any double coset TIOJ] (v, s) T[@‘] f or which the two representations (t, p) -+ AJK (s-l (u-‘tp (u) , s-‘ps),

l-l [OJ’I) (t,P)tT([sw)lnr~‘l~ :;

.~~~(s-‘ps)XK(s1pS)t~~(p-‘(t-1))~~I.(p-1)XK’(p--l). As when we calculated out the criterion for irreducibility, we may expand the matrix element & (8-l (C’tp (a) ) ) , and perform the T summation first. It will give

Now, if OJ and OJ’ belong to different stars, we see that these matrix elements can in no way belong to equivalent representations, and thus the sum will vanish, whatever double coset generator (v, s) we choose. Therefore, the induced representations arising from different stars will certainly be inequivalent. Alternatively, if OJ and OJ’ belong to the same star, then there is a double coset; namely, that generated by (e, c), where ~(0~) = OJ’, for which we may expect that this sum will not vanish. In this case, expression (23) becomes

(24)

x t~~(p-‘(t-‘))~~:.(p-‘)XK’(p-l). Now, & (c-‘PC) = &i (p) , and if we use the matrix element & (p-’ (t-l) ) as tz! (F,’ (1-7 ) = F

F ML(p-7&

(P) t:: (p-l 0-7

),

then we may carry out first the d and u summations to obtain ti: (t-l), and then

the T summation to obtain

Inserting this result in expression (24)) we obtain

SYMMETRY

ADAPTED

FUNCTIONS

OF CRYSTALLOGRAPHIC

GROUPS

‘73

The a- and b-sums remove the Kronecker 8’s, the c-sum yields the unit matrix, while the x-sum yields its trace, e, , Thus we are left with the single term

&,

si&& x”(c-‘P~h”‘w)~

If w’c recall that @ and @’ were not equal but only members of the same star, and thus that xE and xK’ are t,he characters of conjugate representations, since the little groups of these representations arc conjugates of one another, WC WC that this sum reduces to SRKl . Thus IVPhave shown that different stars, or inequivalent representations of the little groups yield inequivalent induced representations of G, while representations belonging to the same star and equivalent representations of the little groups yield equivalent representations of G. COSCLUSION

By deriving a canonical form for the irreducible representations of a semidircact product, in terms of the irreducible representations of its normal subgroup and of thrir little groups, we have attained a goal already reached for direct product groups, whose irreducible representations are very easily constructed in terms of those of their factors. Although the present theory is but a generalization of that. for direct products, the construction is perhaps an order of magnitude more difficult. Moreover, it is yet incomplrtc in the one respect, that a thorough t8nxatment, has not been given of the possible oc’currencae of projective repro’scnt8at,ions, an cxtensivc subject which deserves a separatr treatment. This paper was conceived during a visit by the author to the University of Uppsala, and it is a pleasure to thank both Professor Kai Siegbahn for the hospitality of the Fysikunn and Professor Per-Olov LGwdin for the hospitality of the Quantum Chemistry Group, as well as the latter for a number of very interest,ing discussions. The author also thanks Professor M. A. Melvin for many discussions regarding his method of factoring projection operators and I)r. Charles Herzfeld for his interest, and encouragement’. The partial support of t,he linited States Air Force during the w-riting of this paper is also gratefully acknowledged

I .” 3.

4. 5. 6. 7. 8.

’ li:~-mm WI(;NER, “ Crruppentheorie und ihre Anwendung auf der Quantenmechanjk Atomspektern,” pp. 120-133. Vieweg & Sohn, Braunschweig, 1931. .J. RIID NIELSEN AND L. H. BERRYMAX, J. ChenL. Phys. 17, 659462 (1949). T. VENKATARAYTTD~,Proc. Indian itcad. SC?. A17, 75-78 (1943). ,JOHNE. KILPATRICK, J. Chem. Phus. 16,749-757 (1948). A. 11. FOKKER, Physica 7, 385-112 (1910). M. .4. MEI,VIN, Rem. Modern Phys. 28, 18-44 (1956). P. 0. Liiwnrx, Suppl. Phil. Msg. 5, l-171 (1956), Article 3. Hre B. L. VAN DER WAERUEN, “( Iruppen von Linearen Transformationen,” Springer, Berlin, 1935.

de1

pp. 55-78.

T1

MCINTOSH

9. GEORGE W. MACKEY, Am. J. Math. 73, 576-592 (1951). 10. H. BOERNER, “Darstellungen von Gruppen,” pp. 86-91. Springer, BerIin, 1955. 11. R. BRAUER, “Uber die Darstellung der Drehungsgruppe durch Gruppen linearer Substitutionen,” dissertation, Berlin, 1925. 12. G. FROBENIUS, Sitzungsber. Preuss. Akad. p. 303 (1901). 13. I. SCHUR, Sitzungsber. Preuss. Akad. (l), pp. 164-184 (1906). 14. F. SEITZ, Ann. Math. 37,17-28 (1936). 15. GEORG WINTGEN, Math. Ann. 118,195-215 (1941). 16. GEORGE F. KOSTER, Notes on Group Theory, Technical Report 18 (1956) Solid State and Molecular Theory Group, Massachusetts Institute of Technology, pp. 159-179. 17. GEORGE W. MACKEY, Proc. Natl. Acad. Sci. 0’. S. 36, 537-545 (1949); Ann. Math. 66, 101-139 (1952). 18. H. V. MCINTOSH, The Representation Theory for Finite Groups, RIAS Technical Report 57-4 (1957) (unpublished). 19. See GEORGE A. MACKEY, “The Theory of Group Representations, Part III,” University of Chicago lecture notes, Summer, 1955: dcta Math. 99,265-311 (1958). 20. H. BETHE, Ann. Physik 3,133-208 (1929). 21. R. J. ELLIOTT, Phys. Rev. 96, 280-287 (1954). 82. W. OPECHOWSKI, Physica 7, 552-564 (1940). 23. L. P. BOUCHAERT, R. SMOLUCKOWSICI,AND E. WIGNER, Phys. Rev. 60,58-67 (1936). 24. JOHN S. LOMONT, “Applications of Finite Groups. ” Academic Press, New York, 1959. 26. J. ZAK, J. Math. Phys. 1, 165171 (1960). 26. H. V. MCINTOSH, J. Mol. Spectroscopy 6,269-283 (1960).