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 deter- mined by the algebra of representation of their indecomposable compo- nents.] 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.representa- tion, 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 repre- sentation 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 repre- sentation 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 irre- ducible, 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 re- ducible." 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 repre- sentations 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 irre- ducible 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 auto- morphism 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,.