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CHAPTER I INTRODUCTION A. Motivation and Scope of Work The concept of symmetry is central in both geometry and physics, and group theory provides1 the most appropriate language for the discussion of symmetry. The symmetric groups S, and the alternating groups A,, corresponding to low values of the positive integer n arise directly in a number of specific geometric and crystallographic problems. These groups are also important for a quite different reason. Since it is easy to calculate with the group elements themselves and to obtain explicit representation matrices, these groups serve both to. illustrate the general theory and to suggest conjectures. It was in the latter spirit that the authors computed for their own use most of the information presented in the tables. It is our hope that this material may be of value to a wider audience. In the summer of 1961, when much of this work was done, two of the authors were interested in extending the methods of the quantum theory of anguIar momentum to an arbitrary group. To this end we studied the algebra of representations of the cyclic, dihedral, and quaternion groups and also the algebra of representations of each of the non-abelian groups of order less than 32, as listed for example in the text of Coxeter and M~ser.~Although the problem of "simple reducibility" has been solved314 to our satisfaction, the detailed collection of facts about these special groups is of interest for other reasons. From the mathematical point of view the special examples serve to illuminate the theory of the semidirect product and of induced representations and to suggest conjectures con- cerning, for example, the algebra of representations of a semidirect product in terms of the algebra of representations of the original group. In physics many of these groups are important for solid state the~ry.~We believe also that these considerations have relevance for the theory of the quantum- degree-of-freedom groups (QF,) as introduced by Weyl and Schwinger." Understanding of this physical context is, however, quite unnecessary for using the tables. The plan of this paper is then as follows. The main facts about the special groups considered are presented in tabular form in Chapter I1 (the cyclic, dihedral, dicyclic and QF, groups) and Chapter 111 (all other non-abelian, (1) 2 RICE UNIVERSITY STUDIES non-decomposable groups of order less than 32). [Groups which are direct products of included groups (decomposable groups) are omitted since their algebra of representations is completely and explicitly determined by the algebra of representation of their indecomposable components.] The reader is assumed familiar with the basic facts of group representation theory; for details he is referred on the physical side to the books of Wigner, and Hamermesh7; on the mathematical side to the books of Curtis and Reiner, and Robinson.Wowever, for convenience, some main facts about the algebra of representations, semidirect products, and induced representations are collected in Sections B, C, and D of this introductory chapter. Section E contains only a few remarks on the way in which the tables were prepared. We have made the tables as explicit as possible; however, for reasons of space it was necessary to adopt some conventions, and these are explained in Section E. Finally the table of Chapter IV summarizes the essential facts about the groups considered and serves as an index to the tables. B. The Algebra of Representationsi~!' Let G be a finite group. We.consider the irreducible representations of G over the field K of complex numbers and label the equivalence classes of irreducible representations as r, . .. r,,,. The equivalence classes of reducible representations may be considered as linear combinations of I?, with non-negative integral coefficients. The Kronecker product of these irreducible representations-rl. and r,,for example-is again a.representation, which we write rl,x I'!. r,, x rl is In general reducible, i.e., r, X rl m 1 I = Z CI,, r,. The "structure constants" Cl, are non-negative integers. r=l The associative system of classes of irreducible representations of G (with Kronecker product multiplication) we call the "algebra of representations" of G. Note that it is not strictly an algebra over the ring of integers unless we extend the system to allow subtraction of irreducible representations. The multiplication table for the irreducible representations for each group considered is given explicitly in the tables of Chapters I1 and 111. Irreducible representations may be classified by the value of the Fro- 1 benius-Schur invariant CT., defined by CI- = y Z x~(R?', where it is the order of G and the summation is extended over all group elements R. X1'(R) is the character of the group element R in the irreducible representation r, i.e., the trace of the matrix corresponding to R. Irreducible representations which may be transformed to real form have Cr = 1; those with real characters for all R but which cannot be transformed to real form have Cr = - 1; those with con~plexcharacters have Cr = 0. THE ALGEBRA OF REPRESENTATIONS 3

Representations with Cr = 1 are sometimes called integer, those with r = - 1 half-integer. [An abelian group has no irreducible half-integer representations.] The values of Cr are included in the tables for each irreducible representation of each group considered. Given an irreducible representation r of the finite group G, the Kro- necker square r X r may be decomposed into its symmetrical and anti- symmetrical parts, which we call r @ r and r O r respectively. The corresponding character formulas are:

xrOr (R) = ?h [+f(R)I3 + ?h [+I' (R2)], xrOr (R) = ?h (R)]" $5 [+r (R"] . An irreducible representation which occurs as a constituent of some I' O r with r integer and irreducible, or as a constituent of some r O r with I' half-integer and irreducible, is called even. An irreducible representation which occurs as a constituent of some r Or with r half-integer and irreducible, or as a constituent of some r O r with r integer and irreducible, is called odd. Even and odd representations are indicated in the same table under the heading "algebra of representations" for each group by the letters e and u respectively. Tnteger representations which are neither even nor odd are indicated in the same table by the letter n. f A group G is said to be multiplicity free if no C,, exceeds one. G is said to be urnbivalent if every element is in the same class as its reciprocal. For an ambivalent group no irreducible representations have Cr = 0. A group which is both multiplicity free and ambivalent is said to be "simply reducible." For a simply reducible group even and odd representations are necessarily integer and no representation is both even and odd. The termi- nology, of course, is suggested by the usual names for the representations of the group of rotations in three dimensional space (including spin representations). If a group M is the direct product of two groups G and H, the algebra of representations of M is completely determined by the algebra of representations of the two groups G and H. For this reason the direct product groups are not included in the tables of Chapter 111, although for com- pleteness they are included in the summary table of Chapter IV. The irreducible representations of M can be obtained by taking all Kronecker products of irreducible representations of G with irreducible representations of H. The "structure constants" of M are obtained by ~~lultiplyingstructure constants of G by structure constants of H, and the Frobenius-Schur invariant of a representation of M is obtained by multiplying the Frobenius- Schur invariants of the appropriate representations of G and H. Further, the Kronecker product of an even representation of G with an even (odd) representation of H is an even (odd) representation of M and the Kro-

THE ALGEBRA OF REPRESENTATIONS 5

(1) If u maps H into the group of inner automorphisms of G and the center of G contains the unit element only, then G 6H3 G x H' with Hi = H. (2) If L is a subgroup of F, the group of automorphisms of G, then the natural semidirect product G 8L is a subgroup of the holomorph G 8 F. If L is normal in F, then G6L is normal in G 6 F. (3) If L and M are subgroups of F with only the unit automorphism in common and if each automorphism of L commutes with each automorphism of M, then (G @ L) O M = (G @ M) @ L = G @ (L x M) where 012 means rrl a2 (first u2 then u~). D. Induced Reoresentations The characters of the groups presented in the tables were calculated by induction from the representation of a subgroup. To a given irreducible representation of a subgroup G of a group J the induction process asso- ciates a representation of J which may be reducible. For the general definition and properties of induced representations the reader is referred to the books of Curtis and Reiner,8 RobinsonQr Lomont.ll In this section we record only special cases of the general theory. If G is a normal subgroup of J, two representations r, and r2of G are said to be conjugate with respect to J if there is an element R in J such that Tl(a) = r2(R1a R) for all a in G. r,(a) means, of course, the matrix representing the group element a in a fixed representation belonging to the equivalence class r,. A maximal set of inequivalent irreducibIe representations of G which are mutually conjugate with respect to J is called an orbit of G with respect to J. Representations of G which are in the same orbit lead to equivalent induced representations of J. The representation of J induced by the representation r of G is irreducible if and only if the number of distinct irreducible representations in the orbit containing r (this number is called the order. of the orbit of r) is equal to the order of J divided by the order of G. The last statement is a special case of a theorem of Mackey.I2 Finally we make some remarks about the special case when J is the semidirect product of an abelian group G with the cyclic group of order two. Let T be a representation of G with character (D, r1the induced representation of J with character x. Explicitly one has: X(a,x) = @(a)t 3)(a, (a) ). If r is self-conjugate (i.e., belongs to an orbit of order one) then r' is reducible and has two inequivalent one-dimensional irreducible con- stituents and these are integer or not integer (i.e., have Cr = 0) according to whether r is integer or not (r cannot be half-integer since G is abelian), If r belongs to an orbit of order two, r1 is irreducible and of degree two; r1 is integer if and only if the two representations of the orbit of r are 6 RICE UNIVERSITY STUDIES

either both integer or the complex conjugates of each other. For the case J = G O C2 it is also possible to obtain explicit results for the algebra of representations; however, they are too complicated to present here. E. Preparation of the Tables The preparation of the tables does not require detailed discussion. All calculations were done by hand without the assistance of electronic com- puters. Classes and normal subgroups were found by a systematic search procedure. Although many of the character tables given are available in the literature, they were all computed anew by induction from representations of suitable subgroups. The multiplication table for the algebra of representations, the Frobenius-Schur invariant, and the symmetric and anti- symmetric Kronecker squares were then determined'by simple calculations with the characters. To find the automorphisms, we tested transformations of the group elements systematicaIly to find which one preserved the defining relations of the group. Once the automorphisms were known, their structure was determined essentially by trial and error. AIthough little labor was needed in connection with the algebra of representations, it would not seem practicable to extend our work on the automorphisms to larger groups without more systematic methods or electronic aid. In the case of C,, and D,, the tabular material has been supplemented by some textual discussion to illustrate the nature of the arguments used in each case. F. Explanation of Tables In the tables we have tried to assemble systen~aticallyin one place the main facts about the structure of the groups in question and their representations. This means inevitably that there is duplication of published material and that some tables contain only trivia. We do not claim that much of the material is original, only that this is a useful compilation. It is hoped that the tables can be used with only a rather superficial understanding of representation theory. Direct product groups are excluded for the reason discussed in Section B above. All other groups of order less than 32 are included. The cyclic, dihedral, and dicyclic groups are treated as families in Chapter 11, the others as individuals in Chapter 111. The summary table of Chapter IV provides a convenient index. Each numbered section of Chapter I11 corresponds to one group J. Each group is given a name, sometimes the name is taken from Coxeter and Mo~er,~sometimes it is just that of a semidirect product decomposition, and in one case (the "Pauli group") it comes from usage in physics. As part of the heading for each group we list also the order and the Coxeter and Moser notations, although such notations are not explained here. The THE ALGEBRA OF REPRESENTATIONS 7 defining relations (generators) are taken from Coxeter and Moser" usually the form was selected that sacrifices elegance to ease of manipulation. Letters R, S. . . are used systematically for the generators; E is, of course, the unit element of the group. The table of classes seems largely self-explanatory, Each column contains a list of the elements in one class. The classes are given arbitrary names in the first row for use of reference in the character tables. By the "order" of a class we mean the number of elements in it, by the "period" we mean the order of an element R of the class, i.e., the least positive integer rn such that Rnl = E. The table of normal subgroups of J requires more discussion. The first column lists the generators of the normal subgroups, the second the abstract group isomorphic to the ~ormalsubgroup in question. The notation (d, m) is used in some tables for the greatest common divisor of the integers land m. If there is a family of conjugate normal subgroups, only one in general is listed although a note below the table will call the reader's attention to the existence of conjugate unlisted subgroups. The third column lists the abstract structure of the factor groups H =- J/G. In the fourth column an indication is given as to whether or not there is a realization of H in J such that G and H have onIy the identity in common. If there is such a realization, the generators are given; if no such realization exists the word "No" is written. The notation {R, S1 means, as usual, the subgroup generated by R and S. Finally, the last column indicates the definition of the mapping (7 in the event that J = G6H. A dash in the last column indicates that J cannot be written as any such semidirect product. If H has a single generator T, the automorphism of G listed in this column is n (T). Beneath the table of normal subgroups are given the center, commutator subgroup, a composition series of J, and the factor groups of the composition series. CHAPTER 11 SOME FAMILIES OF GROUPS

A. The Cyclic Group CN of Order N Generators: R" = E.

Direct Decompositions: CNis indecomposable if N is a power of a prime. If N = LM and L and M are relatively prime then CN = CL X Cx. Any finite abelian group is a direct product of cyclic groups of prime power order.

Classes: Since CN is abelian, it has N classes each consisting of one element only.

Normal Subgroups: All subgroups are normal. If N is prime, then Cs contains no proper normal subgroups and is thus simple, Each element of Cs has then order N. If N = LM, then the group generated by R" is a normal subgroup of order M and similarly the group generated by RM is a normal subgroup of order L. If (L, M) = q P 1 then RT,and Rhlalso generate these normal subgroups CaI and CI,, but since (L, M) + 1, these subgroups have more than the unit element in common so that there is no direct decomposition CX = Cr, X Car. It is clear that Cx, as all abelian groups, cannot have semidirect decompositions that are not direct. The center of CN is just Cs itself. The commutator subgroup of any abelian group, and thus Cs, is just the unit element, CS is solvable and its composition series contains just cyclic groups.

A utomorphisms: C, has, of course, only the identity automorphism as inner automorphism. For a prime p, the automorphism group of C, is just C,-,. The automorphism group of C,a, where p is an odd prime, is THE ALGEBRA OF REPRESENTATIONS 9

Ccp-l)pa-l,and the automorphism group of C30( is C2a-2 x C3. If N = LM where (L, M) = 1 then the automorphism group of Cx is just the direct product of the automorphism groups of Cr, and CJI. If (L, M) = q # 1 then the automorphism group of Cs is just the direct product of the automorphism groups of Cordand CJI,, (where qL # N and M/q + 1). Characters: All irreducible representations are one-dimensional and there are as many representations as there are elements. The characters and the representation matrices coincide and the characters are elements of an abelian character group which is isomorphic to Cs itself. We can define the irreducible representation r = j by Xi (A) = ri = eii(b 2.x where r is a primitive Nth root of unity and d, = -(j = 0, I, 2, . N - 1 ) . N .. Algebra of Representations: With the Kronecker product as the Iaw of multiplication the irreducible representations of Cs are just the elements of a group isomorphic to CN itself. Explicitly one has

xi (A) x3 (A) = Xt+-' (A) where i + j is reduced modulo N. Thus the characters yield an explicit realization of the isomorphism between Cs and the additive group of integers modulo N. If N is odd, only the unit representation is integer and it is also even. If N is even there are two integer representations, i.e., for N = 2m, the unit representation and the representation j = m are integer but the representation m is neither even nor odd. B. Dihedral Group, DN of Order 2N Generators: RN = S2 = (RS)Z = E. Direct Decompositions: D, is indecomposable unless N = 4M + 2 with M integral (so that N is twice an odd integer). We will then have D,,I+:! = C2 X DzaI+l (Cr generated by R2"+" DD2a1+rby R9 and s). Classes: The class structure depends on whether N is even or odd. 10 RICE UNIVERSITY STUDIES

(i) Classes of Dzar

Name E R" R" S RS

R" R.X S, FS, RS, E3S, Elements E Rs.a R4S, . . .Rs3S R5S, . . .RN-IS

Order 12 1 M M Period 1 ma) 2 2 2 a

Name E R" S

R" S, RS Elements /I E R,, R2S, .. . Rr-IS

Order 12 N

Normal Subgroup G Factor Group Direct Conditions in J,, ,, on N Generators Structure H - Dv/G J 2- G BH Product? N=2 None R S R -+ R-' c, c? only

N =CNI R"': R -+ R 171 = 2 R' C,,, D, (4m)= 1 S R 4 R1 only

N=Lm R' C,,, D, No - - (l,rrt ) F 1

N = 2 I71 Rz, S Dm c? ~"t R-kR C' X D,,, rrt = odd

R" R' N=2m R:, S D,,, C, AB S + RAS THE ALGEBRA OF REPRESENTATIONS

Center: N=2m. Center(R?7z)=CCz,m#1. N = 2. Center = D,. N = 2m + 1. Center = E. Commutator subgroup : N=2m. C,,,={Rsj. N=2m-t- 1. C2,1L+l= {R], Composition series: DN has a maximal subgroup CX, and thus one composition series for Ds contains only cyclic groups. A utornorl~hisms: Dx admits the automorphism A which takes R into R and S into RS. Since Ax = E, we see that A generates CK.If S, is any automorphism of Cx then S,, is also an automorphism of Ds.We see then that the automorphism group of Ds is just the holomorph of Cs, i.e., the natural semidirect product of C, with the automorphism group of CN. For D,, there is an added automorphisrn which interchanges R and S and the automorphism group of D2 is D.{ generated by U which takes R into RS and S into R and by V which takes R into R and S into RS. From the table of classes it is clear that the automorphism A is inner only if N is odd. However, AS and S,,., are always inner and may be taken as the generators of the group of inner automorphisms. Thus for D,,,, the group of inner automorphisms is (AYm = (S,,-1)" E, S,,.cl A2SllTl= k2, which is D ,,,, and for D2,,r+lthe group of inner automorphisms has the definition (A")"~z-+l = (S,,.1)2 = E which is D2,,+l. Of course, since Ds is abelian, the identity automorphism is the only inner automorphism. The inner automorphism group of DiI is D:! itself, which is in fact the entire automorphism group; in other words, D,

Class 11 E R" (1 < < m) , s

2 cos CY j 4

lo and l1 could be obtained from the reducible representation induced by j = 0 of Callt+l. ,

D2,,P Class 11 E ~"(1

2 cos CY j 4 i 2 2cos jT 0 0 (l

Algebra of Representations: D2nL + 1 Character group: ( 1 ) = C2

It is useful to note expIicitly the rule that: j x j = (j + j') @ / (j- j') 1, where j t j' 3 2m f 1 - j - j' if j + j' > m. Moreover: 0 3 lo@ ll.

r dimension Cry even-odd r@r r@r RICE UNIVERSITY STUDIES THE ALGEBRA OF REPRESENTATIONS

D2,,,(cont'd.) Case I. m = 2k + 1 - r / dimension Cr even-odd

... .*. ... . , ......

e k 2 1 (0,O) @ 2k (0,l) k-even k-odd RICE UNIVERSITY STUDIES

D2,,*(cont'd. )

Case 11. m = 2k + 2

rmension Cr even-odd r@r rOr

e n 1 (0,O) @ 2k II k-even k-odd n 1 k-even k-odd" W>O)@ (190) @ (1,l) (0,l)

e n k+2 2 1 (0,O) 63 2k 11 k-even k-odd

Note that D,, is always ambivalent and multiplicity free. THE ALGEBRA OF REPRESENTATIONS

C. The Dicyclic Group QN of Order 4N [< 2,2, N > in Coxeter's notation.] Q, = Ca is abelian and is explicitly excluded from what follows. Q2 is the famous quaternion group; because of this, we have chosen the notation Q~ to symbolize the family of dicyclic groups. Generators: R2" = E, S3 = Rx, SIRS = R-l or, alternatively RK = SG(RS)?. Direct Decompositions: QN is indecomposable. Classes: If we identify the generators R and S of Q, with the generators R and S of D~Nwe see that the conjugation rules S'l RaS = R-" and R-l SR = SR2 hold equally in QN and DgN.The elements of each group can hence all be expressed, using the conjugation rules, in the form Ra or RaS. With this identification the class structures of DgNand QN are identical, i.e., the "same" class contains the "same" elements. Of course in Dx the elements of the classes S and RS have period 2 while in QN the elements in the classes S and RS have period 4. Just as for D2~,QN has N + 3 classes.

Name 11 E Ra(=1,2..N-1 Rs S RS Elements E R", R2S.m RS S, R, S, .. . Ry'2 S RS, R3S, .. . RX'l S

Order 1 2 1 N N

Period /( 1

Normal Subgroups:

Cond~tions Normal Subgroup G Factor Group in 57 CT Direct on N Generators Structure H - Qy/G Q, = GBH Product

None R C?X cz - A -

N even R? C\ Dz - -

N odd RZ c\ c4 S S:R'+RR? N=l

None RN C2 Ds - - -

N =Lm R ' c2sl D, - - -

N = 2tn R"S Q,,, C, - - - 18 RICE UNIVERSITY STUDIES

Center: {Rr] e C2. commutator subgroups: (R2} CS. Composition series : Since Qs 3 CZs, all Qu are solvable groups. For N even the compo- C, sition series may begin in an additional way: QS 3 Qs,~ 3 Cs. C2 C:! Note that Q2 is the snlallest hrrmiltoizian group, that is, it is the smallest non-abelian group all of whose subgroups are normal. In general, Qx is not hamiltonian. Automorphisms: If A maps R into R and S into RS and S,, is the automorphism of Czx that takes R into RL, then A and S,, are automorphisms of QS and in general A and S, are generators of the automorphism group with appropriate choice of k. We note that A" = E and SISL1AS/,= All. Thus the automorphism group is just the automorphism group of Da~,i.e., the h010- morph of CZS. For the special case of Q2, R + S is an independent automorphism, and the automorphism group is S4. The inner automorphism group is generated by A' and S,,-1. Thus the group of inner automorphisms of Qs is Ds for N 3 2. Characters: The representations of Qs may be found in very much the same way as for Das earlier. The one-dimensional representations may be lifted from the representations of the factor group Qx/center. This factor group is D2 for N even and C4 for N odd, and these two cases for Qs must hence be considered separately. The two-dimensional irreducible representations -for N either even or odd - may be obtained by induction from the representations of Czx having an orbit of two components. These are the representations e""'s with j = 1,2,. .. N - 1. This completely determines all the irreducible representations of Qs, and leads to the character tables that follow. Qy : N even

--[ERG = 1, ... N - I R\ s RS

(- I)* 1 -1 1

2 cos (=,-/N) 2 cosl- 0 0 (I= I, ... N- I) THE ALGEBRA OF REPRESENTATIONS

QN : N odd

Algebra of Representations: Since the character tables are identical for Qx and DaN,if N is even, it follows that the algebra of representations is also the same for both. Thus we find (first table of this group on following page) : QN : N even

r dimension Cr even-odd r@r r@r

(020) 1 1 e (0,O) -

(0,I) I 1 I4 (0.0) -

eN-Omod4 N rrNs2mod4 0 @ 0 1 (%I) j(j= T) 2 odd - 0,@ 1,)@ 1,) (0) \- 1

Note: 2j = 2(N - j) if 2j > N - 1. RICE UNIVERSITY STUDIES THE ALGEBRA OF REPRESENTATIONS 21 RICE UNIVERSITY STUDIES

Qx : N odd

r dimension Cr even-odd r@r r O I'

(0) 1 1 e (0) -

(1) 1 0 (2) -

(2) 1 1 U (0) 7

(3 1 0 - (2) -

i (i odd) 2 - 1 - (2) @2j (0) - e j-Omod4 j (j even) 2 uj52mod4 (0) @2i (2)

Remark: These tables show that Qs is always multiplicity free. Since QN is ambivalent only for N = even, it follows that QN is simply reducible for N = even. The contrast with Das (which was simply reducible for all N) should be noted. Particularly striking is the fact that although DP~and QN (N even) have the same character table, the representations differ appreciably in their properties (for example, Cr = 1 for DPx but Cr = * 1 for QN, N even). D. The Quantum Degree of Freedom Group: QF, Structure: QF, = (C, x C,) 6 C,, p = prime > 2 (a: T + T, R + TR, see below). Order: p3 Generators: R71= Sa = TI]= E, RS = TSR +Rm Sn = TmnSn Rm. ST = TS Classes: Every element of the QF, may be put in the form: g = TaRbScwhere a,b,c, are the integers 0,1,2,... p- 1. THE ALGEBRA OF REPRESENTATIONS 2 3

The elements (T) = C, beIong to the center and yield p classes of one eIement each. The elements g = TaRbSC, b -I- c f. 0 form a class for each b, c of p elements. Hence there exist p2 + p - 1 classes.

Ta Rb SC Name E [b=0,1, p-l;c=O,l, p-11 [a= 1,2, ...p- 11 ...... b+c#O

Elements 11 E Ta (E,T,T2,.. . Tp-l) RbSc - - Order 1 1 P

Period 1 P P

Note that this tabIe would be wrong for p = 2, D,.

Normal Subgroups:

Normal Subgroup G Factor Group 0- Generators Structure H = J/G HinJ? J-G@H

The normal subgroup C, X Cp may be generated by many equivalent choices; C, x Cp= {Tf x {RbSC)b-t- c#0. Center: {T)=c,. Commutator subgroup: (T) = C,. Composition series : QF, 3 C, x C, 2 C, 3 E. CP CIJ CIJ (Thus QF, is solvable.) A utomorphisms: Inner: Conjugation by any element has the effect: T + T, R + TaR, S -+ TbS. RICE UNIVERSITY STUDIES

Hence the inner automorphism group is generated by the two elements: A: T+T,R*TR,S-+S. B: T-+T,R+R,S+TS. These obey the relations : A" = B3 -. E; AB = BA. Hence the inner automorphism group for QF, is: 9' = C, X C,. Outer: The complete automorphism group consists of any transformation: T -+ Th, R + T@RnSb,S -+ TJ'RcSfl, satisfying the relation: 1; 1 hmodp,h $ 0, with*, p,r, a, b, c, dintegersmodulo p. The automorphisms modulo the inner automorphisms form a (factor) group: 0 = GL (2,p)-the general linear homogeneous group consisting of all non-singular 2 X 2 matrices modulo p [Coxeter, p. 921. GL (2,p) is of order (p - 1)" (p + 1 ); Coxeter does not give any generators or relations for this group. The center of the group GL (2,p) consists of the group C,.l generated by U: T + Ta, R + R, S + S; Ulj-1 = E, (or is a primitive pt" root). The 2 x 2 matrices (of integers modulo 11) having determinant 1 form a group, SL (2,p) of order p (p" 1). This group is the quotient group GL (2,~)/C,-1. The complete automorphism group contains C, x C, generated by { A,B) ; this is a group of order (p - (p)" (p + 1).

Characters: From the fact that J/ IT) = C, x C, one may "lift" pqrreducible characters from the factor group. The remaining p - 1 irreducible characters may be obtained by induction from the normal subgroup C, x C, generated, for definiteness, by (T) X {R].Denoting the character of IT} x {R) by X(~Ji),[T -+ W], R + o,fc; 1 e-"I"-/* with j = 0,1, . . . 11 - 1; k = 0,1, . . . p - 11, one sees that conjugation by S * X(jJ~)* x(j.j+L).Hence the orbit of x(jJ:' for fixed j(j f 0) consists of the y characters k = 0,1, , . . p - 1. It follows that a representation from each of these (p - 1) independent orbits induces an irreducible character of QF,. Each of the remaining orbits (of one element ea~h-~(~,~))splits into p irreducibIe characters; these are the pvirre- ducible) lifted representations. In this way the following character table is obtained. THE ALGEBRA OF REPRESENTATIONS 25

Ta Class b = 0,1,. .. p - 1; a=1,2,... p-1 c=O,l, ...p-1; Orbit bfc+O of

w = exp (2 T ip'l) .

Algebra of Representations: Character group: (r(o,l))x (I?(,,,,) C, x C,.

r (k<,lO (if 1

Character group (k,l) (i' C, x C,

(j + j') CB . . . CB (j + i') (i) (i) times

Note: (j + j') denotes the representation (d) where j -t j' = /mod p.

r dim. Cr even-odd r~r VO~

22j(2j)e ...@( 2j) (i) P 0 eu -p + 1 times p- - 1 times 2 2

Note: 2j, 2k, 2/, are a11 modulo p. CHAPTER 111

TABLES FOR INDIVIDUAL GROUPS

A. Tetrahedral group, A4: J = D2@ C3 Order: 12 Generators: R3 = S2 = (RS)3 = E.

Classes:

S, RSR2, R, SR, R2,R2S, E1ements 1 R2SR RS,R2SR2 SR2,RSR Order 1 3 4 4

Period [I 1 2 3 3

Normal Subgroups:

Normal Subgroup G Factor group IJ Generators Structure H = J/G J--G@JH

Center: {El. Commutator subgroup: { S, RSR2 ] = Dz. Composition Series : THE ALGEBRA OF REPRESENTATIONS

Automorphisms: Inner: X = ( A,B,} - A, A: R -+ R?3R2, S -+ RSR2; A3 = E. B: R + R2SR2,S -+ S; B3 = E. (AB)3= E Outer: X: R+ R< S-+ S; X2= Ed B = (XA)"AX)" (XA)" E. Automorphism group: d = {A,X 1 = S, Characters (induced from D3) :

Algebra of Representations: Character group: C:, = { (rl, ) .

I? 1 dim. Cr even-odd r@r r8r RICE UNIVERSITY STUDIES

B. J= C8@C2.(J= (2, 2 j 2)of Coxeter) Order: 16 Generators: RS=S2=E, SRSzRj. Classes:

Name ERR" R" R1 R6 S RS R2S R3S

-- R, R2 R", Rq6 S, RS, R3S, R3S, Elements E Rz 11 Ri R% RS RFS R7S

-- - Order (1 i 2 1 2 1 1 2 2 2 2

Period 1848242 8 4 8

Note that this group is not ambivalent. Normal Subgroups:

Normal Subgroup G Factor Group IJ Generators Structure H-J/G J~FBH

Conjugate normal subgroups can be obtained from these listed in the table by application of the automorphism R + RS, S -+ S. Center: (R" = c*. Commutator subgroup: {R" = Cz. Composition series: J 2 C, 3 C* 2 C:! 3 E. C2 Cr C2 C:! THE ALGEBRA OF REPRESENTATIONS

A utomorphisms: Inner: 9= {A,B } = Dn. A: R+R\S+S; A'kE. B: R + R, S + R4S; B" E. AB = BA Outer: X: R + R3, S 4 S; X2 = E, XA = AX, XB = BX. Y: R+RS,S+S; YL=E, YA = AY, YB = ABY, XY = YX. Automorphism' group: dg (D:!XC2)@C2. (v:A+A,B+AB,X+X). Characters:

Class E R RM3 R4 Re S RS RS RJS Orbit of 1212112222CR

0 lll1llllll 0

1 1 1 1 1 1 1-1 -1 -1 -1 0

Algebra of Representations: Character group: { 4 x { 1} = C4 x C2. RICE UNIVERSITY STUDIES

r dim. Cr even-odd rOr r@r

0 1 1 e 0 -

1 I 1 n 0 -

2 1 1 n 0 -

3 1 1 n 0 -

4 1 0 - 2 -

5 1 0 - 2 -

6 1 0 - 2 -

7 1 0 2 -

8 2 0 - 4@5@6 7

9 2 0 4@5@7 6

C. J = C8 O C2 (J = ( - 2,4 / 2 > of Coxeter) Order: 16 Generators: Rs = S2 = E, SRS = Ra, Classes:

Name I/ E R RV4 R5

R, R" R5 , S, R?S, RS,R3S Fiementi E R3 1 RG' R4 R7 R4S,RGS RS,R'S I I Order 12 2 1 2 4 4 II Period 18 4 2 8 2 4 THE ALGEBRA OF REPRESENTATIONS

Normal Subgroups:

Normal Subgroup G Factor Group Generators Structure H = J/G J-i6H R cs C2 is) R-+R3 R? C, D2 NO -

RS, R2 Qz C2 (SJ RS-+R3S,S2+Sa

Center: {R4{- Cz. Commutator subgroup: {R2]= c*. Composition series : J C8 C4 C2 E. C2 C2 C" C2 Automorphisms: Inner: 9 = (A,B ] - D4. A: R+R3,S-+S; A2=E. B: R+ R, S+ R"; B4= E, (AB)2 = E. Outer: X: R -+ R,S + S; X2 = E, AX = XA, BX = XB. Automorphism group: d = (A,B,X1 = D4 x Cg. Characters: 32 RICE UNIVERSITY STUDIES

Algebra of Representations: Character group: { 1 ) x { 2 ] = D2.

This group is multiplicity free but not ambivalent.

dim. Cr even-odd r@r rOr

1 1 e 0 -

D. J = C4 O Cf. (J = (2,2 / 4;2) of Coxeter) Order: 16 Genemtors: RbsS-'=E, S'lRS=w. Classes:

Name I/E R R2 S S S" RS RS2 RS3 R"S2

R, S, S" RS, RS" RS', K,S? 'lernents R.? R' R2S S2 R2S3 R?S R3SY R3S3 II Order 12 12 12 2 2 2 1

Period 1) 1 4 2 4 2 4 4 4 4 2 THE ALGEBRA OF REPRESENTATIONS 33

Normal subgroups:

Normal Subgroup G Factor Group u Generators Structure H=J/G J? J~C~H

S" RR" D:, D:, NO -

Conjugate normal subgroups can be obtained from those listed in the table by application of the automorphism R -+ R, S -+ RS.

Center: {R"SR")sDz. Commutator subgroup : {R" = c:,. Composition series : J 3 C:, x C, 2 C4 > C2 2 E. C. C:, 2:: C2

Inner:,f = {A,BJ-- D:,. A: R+R,S+R%; A=E. B: R+ R:', SSS; B"E, AB = BA. 0uter:X: R-+RS2,S-+RS; XL=E. Y: R+ RS" S s RR"S; Y" E, YB=BY;YXY=AX:'; BXB=AX=XA, YA = AY. Automorphism group: ,,d= {A, X, Y, B ) = (C, x C1) 6 (C, x C,). (cry: A 3 A, X+ AX:$;cq?: A -+ A, X -+ AX). RICE UNIVERSITY STUDIES

Algebra of Representations: Character group: { 2) x { 4 + C4 X C2. THE ALGEBRA OF REPRESENTATIONS

r dim. Cr even-odd r@r r@r

0 1 1 e 0 -

1 1 1 U 0 -

2 1 0 - 1 -

3 1 0 - 1 -

4 1 1 eu 0 -

5 1 1 eu 0 -

6 1 0 - 1 -

7 1 0 - 1 -

8 2 1 n 0@4@5 1

9 2 -1 - 1@4@5 0

E. J = (C4 x C2) 8 Cz (J = (4,4 12,2) of Coxeter) Order: 16 Generators: R* = Sk((RS)" (R-1S)2= E, (R, S2] Cq X C2, u~s.R4 R3S2, S+S. Classes:

Name E R2 S2 R2S2 RS R3S R S R3 S3

RS, R3S, R, S, RS, S3, R2 s2 R2S2 R3S3 RS3 R352 R2C3 RS2 R2S

Order 1111 2 2 2 2 2 2

Period 1222 2 2 4 4 4 4 Normal Subgroups:

Normal Subgroup G Factor Group in ? u Generators Structure H-- J/G J - G6H ",

J, "2 ~2,SS" D2 Da NO - >- - -?f ;2 R2 or SL CL D 1 No 4 'f, R?S2 Xi NO - c2 c"

RS, RSz D2 Dz No -

RS,R'SS" CeXC2XCa CZ NO - THE ALGEBRA OF REPRESENTATIONS

Characters: 3 8 RICE UNIVERSITY STUDIES

Algebra of Representations: Character group: { 5 ) x { 1 ) = C4 X C,.

r dim. Cr even-odd r@r rOr

F. Pauli group, J = (C4 x C2) @ C2 Order: 16 Generators: ReS2 = T3 = E, RST = STR = TRS,(RST, R) C4 X Cs. cr,r: R -+ TRT, RST -+ RST. Classes:

Name E R S T (RST)? RS ST TR RST RTS 4 Elements E :kT s' T' 2 3 RST RTS RSR STS

Order 12 2 2 1 222 1 1

Period 12 2 2 2 4444 4 THE ALGEBRA OF REPRESENTATIONS

Normal Subgroups:

Normal Subgroup G Factor Group H in J? n Generators Structure H=J/G J = G6H

RST c, D2 NO -

(RST) c2 Dz x Cz No -

R, RST R + TRT RST RST or -+ T, RST T + STS RST + RST or S + RSR S, RST RST + RST

Center: {RSTf 5 Cq.

Commutator subgroup: { (RST)" C2.

Composition series :

A utomorphisms:

Inner: 9 = {A,B] -D2. A: R+R,S+RSR,T+RTR. B: R+TRT,S+TST,T+T. A2 = Bk(AB)? = E.

Outer: X: R + S, S + T, T -+ R; X" E. Y: R+S,S+R,T-+T; YkE. Z: R + R, S + S, T + STS; Z" E,

Automorphism group: d' = S, 8 C2, where: S, = 9 6 D:$, {X,Y} = D3, with n: X-' AX = B, X-' BX = AB, YAY = B, and {Z) 3 C2,p: A+ A,B+ B, Y -+ Y. X-+ XA. 40 RICE UNIVERSITY STUDIES

- n d A -! 3n - d m .%SX AAhhh---w- qgidqO/?.;;-@ @ -EiBxg0 z-zczzc5--- 0, 0, ~2 d m - w w lll

m 3 r3dr3r3+3+riGG C IIIII

33dHddd3GG 1111 I

c" 3HdHdH3+00 $ I I I I

d33ddd3dOO % IIII

dddddt..(3100 2 II II

N A s ,-,3H-3dd-lmm d I I V

t-c ,+d,+3,+U3dOO 1 I I I

d33d3.-lHdOO ZA I I I 1

C.4 ~H~HHHHWOO d I I I I

mw 4dddHdHHc"c"

3 OHmmTf'n\oF00~ U THE ALGEBRA OF REPRESENTATIONS

Algebra of Representations: Character group: (If X {2) X (41 -Cz X Cz x Co,

G. J = (Ca X C:,) @ C2 (J = ((3,3,3;2)) of Coxeter) Order: 18 Generators:

R2 = S2 = Tz = (RST)2 = (RS)3 = (RT)3 = E, {RS, RT) C:i X C:l, cI1:RS -+ SR, RT -+ TR. 42 RICE UNIVERSITY STUDIES

Classes:

Name // E RS RT ST RSTS R RS, RT, ST, RSTS, R, S, T, RST, RTS, 1 SR TR TS RSRT SRT, RSR, RTR, STS Order 11 1 2 2 2 2 9 -- Period 11 1 3 3 3 3

Note that this group is ambivaIent. Normal Subgroups:

Normal Subgroup G Factor Group in J? u Generators Structure H J/G J = G6H

RS RT, T* RS -+ SR or or or RT c3 D3 RS, S RT+ TR or or or ST SR, R ST -+TS

RS, RT R RS + SR RT + TR or or RS 4 SR RS, ST C3 X C:, C, S ST -+ TS or or RT+ TR RT, ST T ST -+ TS

u . RSTS + RSTS RSTS C:c RT, S** Ill'' D:i cr s'. RSTS -+ RSRT q'When RS generates CR,we can have D:< generated by RT, T, by RT, S, or by RT, R, and similarly for the others. :%'*WhenRSTS generates Cs, the same holds as for *. Center: IEI. Commutator subgroup : (RSj X {RTj C:i X CQ. Composition series : J 2 CS X CX2 c:<2 E. C, C:r c:< THE ALGEBRA OF REPRESENTATIONS

A utornorphisms: Inner: 9 = {A,B,Cj = J. A: R-+R,S-+RSR,T+RTR; A2=E. B: R -+ RSR, S -+ S, T -+ STS; B2 = E. C: R -+ RTR, S -+ STS, T + T; CkE. (ABC)' = (AB)3 = (AC)" E. Outer: X: R+ RST, S+ S, T+ R; X4 = B. Y: R+S,S+R,T-+T; Y2=E. The relation (XY):{ = BAC shows that ,d/.f=S4. The relations: Y-lAY = B, Y-ICY = C; X-IBX = B, X-1CX = A together with the other relations above show that A, B, C, X, Y generate d,which is a (rather complicated) group of order 432. Charcrctem: The characters can be obtained from C:r X C;{ by induction. C:i x C:$ has five orbits, four of which induce inequivalent irreducible two-dimensional representations of J, the remaining orbit inducing two inequivalent one-dimensional representations. The character table is thus as follows. 44 RICE UNIVERSITY STUDIES

Algebra of Representations: Character group: ( 1 } - Cz.

r dim. Cr even-odd rgr TO r

0 1 1 e 0 -

1 1 1 U 0 - 2 2 1 e 0@2 1 3 2 1 e 0@3 1 4 2 1 e 0@4 1 5 2 1 e 0@5 1

H. J = Cj O C4 (K-metacyclic; holomorph of C5) Order: 20 Generators: Sj = TkE, TlST = S2. Classes:

Name E T T2 T3 S

T, ST, S2T, T2, ST2, S2T2, T3, ST3, S2T3, S, S2, Elements E S3T, S4T S3T2, S4T2 S3T3,S4T3 S3, S4

Order 1 5 5 5 4

Period 1 4 2 4 5 THE ALGEBRA OF REPRESENTATIONS

Normal Subgroups:

Normal Subgroup G Factor Group u Generators Structure H = J/G HinJ? J=G@H

Center: {El* Commutator subgroup: IS] = C5. Composition series : J 3 C, 3 E. Ca C>

Automorphisms: 1nner:Y = {A,B\ =C50C4. A: S-+S"T+T;A4=E. B: S+S,?'-+ST;B"E. A'lBA = B3 Outer: The only possible outer automorphisms are the operations, X,, defined by : X,: S-+Sa,T+T3.

However, for no value of a does the condition T-l ST = S%ap prop- erly. [It should be noted that the transformation X2 carries the group f S, T) - C; O C, into {Sf,T') G C; O C4. Coxeter does not distinguish these two groups but they are in fact anti-isomorphic. It follows that the operation X2 is not an admissible automorphism] .

Automorphism group: = (C, C4).

The group Cj O C4-the holomorph of Cj-is complete (no outer automorphisms with center consisting of E only). 46 RICE UNIVERSITY STUDIES

Characters:

Algebra of Representations: Character group : { 2 ] = C4.

r dim. CI. even-odd rOr rOr

0 1 1 e 1 -

411 1 err 0@1@[4$4] 2@3@4 THE ALGEBRA OF REPRESENTATIONS

I. J = C7 O C3 (ZS metacyclic) Order: 21 Generators: S7 = T3 = E, T-lST = S2. Classes:

Name E S S3 T T2

9, T, ST, S2T,S3T, T2,ST2, S2T2, S3T2, 'Iements s, ~2,s' 9,s' SJT,S~T, SOT S'T" sSjT2,SOT2

Order 13 3 7 7

Period 11 1 7 7 3 3

Normal Subgroups:

Normal Subgroup G Factor Group u Generators Structure H = J/G in J? J GOH

Center: PI- Commutator subgroup: {S) = Ci. Composition series : J 3 Ci 3 E. C:< Ci Automor[~hisms: Inner: 9 jA,B} =CiOC,. A: S -+ S, T -+ ST; AT=E. B: S-+S4,T+ T; BGE. B-lAB = A4. Outer: X: S -+ S2, T +T; XG= E. B = X'; X-IAX = A. Automorphism group: d = Ci O C6. Note that the group J = C7 O C:

Characters:

Algebra of Representations: Character group: ( 1 } r C3.

dim. Cr even-odd r OF r O I' THE ALGEBRA OF REPRESENTATIONS

J. Symmetric Group on Four Symbols, Si (Octahedral Group) Order: 24 Generators: R* = Sk(RS)" E. Classes:

Name 11 E R S RS R2

Qnts E SR-, RSR, R3SR3 RZSR,S, RSR3, R3SR2S,R2SR2, RSR2S RS,R2SR, R3S, R2SR3,R3SR2 SR, SR3,RSR" RR?,SR'S, 11 R'R''RIS' R2SR2S

Order 1 6 6 8 3

perlod 1 4 2 3 2

Normal Subgroups:

Normal Subgroup G Factor Group u Generators Structure H = J/G J~G~H

Center: E only. Commutator subgroup : {R2,RS) =Ai. Composition series : SI 3 A1 D2 3 C2 3 E. C2 C:' C, C, A utomorphisms: S.l is a complete group, and hence there are no outer automorphisms. The inner automorphisms are generated by A: R -3 R, S -+ R%R, B: R -+ R:'SRr', S -3 S, which satisfy the relations: A4=B2= (AB)3 =E. Hence sf' = 9= Sb. 50 RICE UNIVERSITY STUDIES

Characters:

Algebra of Representations: Character group: { 1) = Cz. - --

r 1 dim. Cr even-odd rg r rOr

Remark: S4 is simply reducible, but the normal subgroup A4 is neither ambivalent nor muItiplicity free. THE ALGEBRA OF REPRESENTATIONS

K. J = (Co x C,) 6 C2 ( (4,6 ( 2,2) of Coxeter) Order: 24 Generators: Ri = SG = (RS)2 = (RlS)2 = E, (S, R"\ - (2, x C, U: S -+ wS5. Clusses:

Name E R2 S S? S3 S6 R2SZ R RS Uemenc- /I RQ S, S2, S3, S5, RR2, R. RS2, RS4, RS, RS3,RSs, RZSG S4 R2S3 R2S R2S4 R3, R3S2, RSS1 R3S, R3SJ,R3S5 I1 Order 112 22 2 2 6 6

Period 1 2 6 3 2 6 6 4 2

Normal Subgroups:

Normal Subgroup G Factor Group u Generators Structure H - J/G HinJ? J=G@H RZ cz Cg @ Cz NO -

S, R2 co x '22 Ca 1 Rs 1 S -+ R2S5

S3, R2 C2XCz {S2,RSj=C30C2 {RS) S3+RZS3

S2,RS, RZ (CQ@ C2) X C2 {S" ) C2 (S3) RS+R2'RS

Center: (R") = C,. Commutator subgroup : {S2)x {Rs)= CZix C2. Composition series :

A utomorphisms: Inner: 9 = {A,Bj -- D,. A: R-+R3S2,S-+S; AG=E. B: R-3 R, S 4 R2S"; B2= E. BAB = A-l, 52 RICE UNIVERSITY STUDIES

Outer: X: R -+R, S + SR'" XX"= E, XA = AX, XB = BX. Automorphism group: d = 9XCX" = {A,B) X (X}. THE ALGEBRA OF REPRESENTATIONS

Algebra of Representations: Character group : { I ] x { 2 ) s D2.

r 1 dim. Cr even-odd rgr rOr

L. J = < - 2, 2, 3 > of Coxeter (ZS-rnetacyclic) Order: 24 Generators: s2= T2 = (ST)" SS = T8 = (ST)12 E. RICE UNIVERSITY STUDIES

Classes:

Name E S2 St SC S7T S3T ST S'T S S' Si S'

S, S.SIT, S'T, ST. S'T, S.StT, I1,SV, S3,T, S7,ST, I/ TS7 TS3 TS TS' TS7T TST TS?T TSST Order 1 1 1 I 1 2 2 223 3 3 3

Period 111 4 2 4 6 3 12 12 8 8 8 8

Normal Subgroups:

Normal Subgroup G Factor Group in ? u Generators Structure H = J/G J = G8H

S" (or SG)

S4 C? QR NO -

ST Cia c2 NO -

S3T c:+ cs is) S3T+ TS"

Center: {S" = c4. Commutator subgroup : {S3TT)= C3. Composition series : J Cl.2 3 C6 3 C:$ 3 E. C:! C, C:! C:, Automorphisms: Inner: 9 = {A,B} A: S + T", T-+ S'I; A" E. B: S+TSiT,T-+T; BkE. (AB)" = E. Outer: X: S -+ S" T -+ T3; XhE. Y: S4Sj, T+T" YY"=E. XA = AX, XB = BX, YA = AY, YB = BY. Automorphism group: s2' = D3 X D2. . (P/? L) dxa = m 56 RICE UNIVERSITY STUDIES

Algebra of Representations: Character' group: ( 6 ] -= C8

r dim. Cr even-odd rOr r Or

0 1 1 e 0 A

M. Binary Tetrahedral Group; (<2, 3, 3> of Coxeter) Order: 24 Generators: R" E, SRS = RSR. THE ALGEBRA OF REPRESENTATIONS

(Two alternative definitions are; (1) A3 = B3 = (AB)" (2) U3 = V2, (U-1V)2= E.) Classes:

Name E (RSI3 R R2 RS (RS)" RSR

RS SR R2S,RSR, SR2, R', S2, E (RS)" :&, (SR)2 (RS)I, (SR)4 R'[RS;~, ~~S~,~~~),R2S(SR)3,RSR(SRI3, S2(RS)3 SR2(SR)S

Order 1 1 4 4 4 4 6

Per~od 1 2 3 3 6 6 4

Normal Subgroups:

Normal Subgroup G Factor Group u Generators Structure H = J/G HinJ? J=G@H

R", RSR Q2 C:< NO

Center: ((RS)3} Cs. Commutator subgroup : (R2S,RSR} s Q,. Composition series : J 3 Q2 3 C2 3 E. C:, C, C,

Automorphisms: Inner: 9 = {A,Bj = A*. A: R + R, S + (RS)" A3 = E. B: R+S,S+R;B"E. (AB)" E. Outer: There is but a single independent outer automorphism, X. X: R+R2,S-+S2;X2=E. X-lAX = A-I; XIBX = B. It follows that the total automorphism group, d , has the structure: &*A4 OCa =S4. RICE UNIVERSITY STUDIES

Characters: w = exp (~i/3)

E Class R RVS RSR (RS)"RS)$ Orbit of ';sd, 1444 6 4 1 Q2 0

Algebra of Representations: Character group: { 1 ) = C3.

r dim. Cr even-odd r@r r~r

0 1 1 e 0 THE ALGEBRA OF REPRESENTATIONS

N. J = C, O Cs Order: 27 Generators: T3 = E, T-IST = S-2, + S, = E. Classes:

Name E S3 SO T T' S S2 TS TS' T'S T?S2

Sn T.TS3. T2.T'S3. S.S4, S', S5. TS. TS', TTTZS,TJS'. TY,,T2S5. 1 TS" T?SV7 SX TS' TSVT?S' T2S8 Order I 1 1 3 3 3 3 3 3 3 3

Normal Subgroups:

Normal Subgroup G Factor Group in ? ~7 Generators Structure H = J/G J=GOH

S CO c3 (TI S -+ si

T, S3 C3 X C3 C.3 NO -

Center: (S3] = cC,. Commutator subgroup :

(S" f -3. Composition series: J 2 C, 3 C3 3 E. c3 C:i C3 Automorphisms: Inner: 9 = {AJX (B) = Cs x Ca. A: T4TS" S + S; A" E, B: T4T,S-+S4; BB"=E. AB = BA Outer: X: T -+ T, S -+ Sy XG= E. B = X2 Y: T-+ T, S-+ TS; YkE. A1 YA = X". X"X:3=X"Y". @ = ,I---sP -d uo:ssn~s!p aas-,,y&ourosr-guy, ayl sey c:a = 1dno.18 ayl ley1 aloN 'z;izX C A 'ZX t zX :sX~ A~XC- A 'Z~c Z~ : alaqM '$a@ ($3x 9)= [ISXI 0 {vll@ [{A1 x Lxll = p :dnol8 ws!yd~ourolnv THE ALGEBRA OF REPRESENTATIONS

Algebra of Representations:

Character group : { 4 1 x { 5 ] = C:, x C, .

r 1 dim. Cr even-odd rgr r@r CHAPTER IV SUMMARY

Summary on the Non-Abelian Finite Groups of Order less than 32:

Order Designation Structure Defining Relzt'tons

R:' = S& E, SRS = R3 6 D:, 5~ S:, C3 O C2 THE ALGEBRA OF REPRESENTATIONS

Integer Reps? Half-integer Ambivalent? mf? QA?% (besides identity) Reps?

Yes No Yes Yes Yes

Yes Yes Yes Yes Yes

Yes No Yes Yes Yes

- - -- Yes No No No Yes

Yes Yes No Yes Yes

Yes No Yes Yes Yes

Yes Yes Yes Yes Yes

Yes No Yes Yes Yes

Yes Yes Yes Yes Yes

Yes No No Yes Yes

"Note: We say that a group is QA (quasi-ambivalent) if it admits an involutary anti- automorphism that preserves the classes. RICE UNIVERSITY STUDIES

Summary on the Non-Abelian Finite Groups of Order less than 32: (continued)

Order Designation Structure Defining Relations

16 (2,214;2) c4@c1 R4=S,=E,SlRS=R" (cont.) - (4,4 1 2,2 ) (C, X C?) 8C? R' = S' = (RS)*= (R'S)" E R2=S2=TZ=E, Pauli group (C4 x C?) 8C, RST = STR = TRS

Do Co 0C2 R" S2= E, SRS = Rs

C2 X D5 ClO 0C.'

22 DIT c,,@ c, R11= S? = E, SRS = RIO THE ALGEBRA OF REPRESENTATIONS

Integer Reps? Half-integer Ambivalent? mf? QA?'. (besides identity) Reps?

Yes Yes No Yes Yes

- Yes No No Yes Yes

Yes No Yes No

Yes No No Yes Yes

Yes No Yes Yes Yes

Yes No No No Yes

Yes Yes No Yes Yes

No No No No Yes

Yes No Yes Yes Yes

Yes No No No Yes

Yes No Yes Yes Yes

Yes No No Yes Yes

Yes Yes No Yes Yes

See note on p. 63. RICE UNIVERSITY STUDIES

Summary on the Non-Abelian Finite Groups of Order less than 32: (continued)

Order Designation Structure Defining Relations

24 Cq X D3 - A (cont.) Dl2 Clz 0C2 ~12= ~2 = E, SRS = ~11

s4 A4 @C2 R4 = S2 = (RS)3 = E

'binary tetrahedral' R3= S3 = (RS)2

(C8 x C2) 0C2 ( 4,6 1 2,2 1 R4 = So = (RS)3 = (R'1S)Z = E or: D2 @I D3 THE ALGEBRA OF REPRESENTATIONS

lnteger Reps? Half-integer Ambivalent? mf? QA?. (besides identity) Reps?

Yes No No Yes Yes

Yes No Yes Yes Yes

Yes Yes No No Yes

Yes No No Yes Yes

Yes Yes No Yes Yes

- - -- Yes Yes Yes Yes Yes

Yes No Yes Yes Yes

No No No No Yes

Yes No Yes Yes Yes

Yes Yes No Yes Yes

Yes No No Yes ?

- Yes No Yes Yes Yes

:See note on p. 63. 68 RICE UNIVERSITY STUDIES REFERENCES 1. Weyl, Hermann. Symmetry. Princeton: Princeton University Press, 1952. 2. Coxeter, H. S. M., and W. 0. J. Moser. Generators and Relations for Discrete Groups. (Ergebnisse der Mathematik, 14, 2nd ed.) Berlin: Springer, 1965. For brevity, this is referred to in the tables as "Coxeter." 3. Biedenharn, L. C., A. Giovannini, and J. D. Louck. "Canonical Defi- nition of Wigner Coefficients in U(n)," Journal of Math. Phys., 8 (1967), 691-700. 4. Derome, J. R., and W. T. Sharp. "Racah Algebra for an Arbitrary Group," Jo~crnalof Math. Phys., 6 (1965), 1584-1590. 5. Koster, G. F. Space Groups and their Representations. New York: Academic Press, 1957. 6. Weyl, Hermann. "Quantenmechanik und Gruppentheorie," ZS. fiir Physik, 46 (1927), 1-46. Schwinger, .J. "Unitary Operator Base," Proc. Natl. Acad. Sc., 46 (1960), 570-579. 7. Wigner, E. P. Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. New York: Academic Press, 1959. Hamermesh, Morton. Group Theory and its Application to Physical Problems. Reading, Mass. : Addison-Wesley, 1962. 8. Robinson, G. de B. Representation Theory of the Symmetric Group, Toronto: University of Toronto Press, 1961. Curtis, C. W., and I. Reiner. Representation Theory of Finite Groups and Associative Algebras. New York: J. Wiley, Inc., 1962. 9. Wigner, E. P. "On Representations of Certain Finite Groups," Am. J. Math., 63 (1941), 57-63. Sharp, W. T. Racah Algebra and Con- traction of Groups. AECL 1098, Chalk River, Ontario, 1960 (un- published). 10. HalI, Marshall, Jr. The Theory of Groups. New York: Macmillan, 1959. 11. Lomont, J. S. Application of Finite Groups. New York: Academic Press, 1959. 12. Mackey, George W. "On Induced Representation of Groups," Am. J. Math., 73 (1951), 576-592. Mackey, George W. "Symmetric and Anti-Symmetric Kronecker Squares and Intertwining Numbers of Induced Representation of Finite Groups," Am. J. Math., 75 (1953), 387-405. 13. Fano, U., and G. Racah. Iri-educible Tensorial Sets. New York: Academic Press, 1959. 14. Hall, Marshall, Jr., and J. K. Senior. The Groups of Order 2"(n56). New York: Macmillan, 1964.