The properties of the groups ON, SON, Sn and An
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
The problems of group theory applied in physics often are reduced to the calculation of the dimensions, the branching rules, the resolution of the Kronecker products, the symmetrized powers and the classification of the irredicuble representations (irreps) for a wide variety of groups. The orthogonal ON and its subgroups, especially the rotation groups SON, symmetric groups Sn and the alternating groups AN have been of special interest to physicists. The n-dimensional rotation groups play an important role in many areas of physics and chemistry. They arise, for example, in the description of symmetrized orbitals in quantum chemistry (Wybourne 1973), in fermion many body theory (Fukutome et al 1977), in boson models of nuclei (Arima and Iachello 1976), grand unified theories (Gell-Mann et al 1978) and in supergravity theories (Cremmer and Julia 1978). Interest in the rotation groups has greatly increased in recent times with study of candidate groups for grand unified theories of the weak, electromagnetic and strong interactions. The group S010 appears to be of particular significance (Fritzsch and Minkowski 1975, Chanowitz et al 1979, Buras et al 1978, Georgi and Nanopoulos, 1979, Witten 1979). The symmetric group has long been of interest to physicists and chemists who have sought to exploit the permutational symmetry associated with many fermion and many-boson systems. For An in physics the isomorphisms ₃ ~ A₃ , T ~ A₄ and I ~ A₅ are well-known in solid state and molecular physics (Lax 1974). The subject of my thesis is devoted to problems concerning dimensions, branching rules and the resolution of the Kronecker products etc for the groups ON, SON, Sn and An. This thesis is organized as follows: In chapter 1 a brief statement is given about the results of the theory of representations of group SN both in ordinary and spin representations that will later be used. On spin representations, I prefer the concept of projective representations to that of the ordinary representation of double groups. Given a brief statement about projective representation I restate the familiar results gotten from double groups from the standpoint of projective representations. In the end of this chapter I present some results relevant to associated and self-associated representations. In chapter 2 I provide some review material on the symmetric functions including basic symmetric functions, Schur functions (S-functions), raising operators etc. In Sec. 2.7 we use the concepts of partitions, frames and numberings to give a simple method to evaluate the outer product of S-functions and the skew S-functions. While in Sec. 2.8 in addition to the known S-function series such as A,B we also give some new series (e.g. R,S ...) and their relations which will be used in the representations of SON. Chapter 3 is devoted to the relation between the symmetric functions and the representations of groups. The relationship between S-functions, Q-functions and the representation theory of groups is outlined. I present the definitions of the inner product of S-functions and Q-functions which play an important role for resolving the Kronecker product of the spin and ordinary irreps of Sn and point out the relation between branching rules, skew S-functions and Q-functions. The results given in chapter 4 are explicit formulae for a complete set of fundamental products from which all possible products of irreps of ON and SON may be evaluated both for n = 2v and for n = 2v ＋ 1. The explicit resolution of the basic Kronecker squares into their symmetric and antisymmetric parts is then given, followed by a complete resolution of the Kronecker cubes of the basic spin irreps of SO2v1‚ and SO2v together with a prescription for analysing explicitly the Kronecker fourth powers of these irreps. These results permit analysis of the Kronecker second, third and fourth powers of any irreps (spin or tensor) of the groups SO2v＋1‚ and SO2v to be made unambiguously. These results are given in a general form that is essentially independent of N, the dimension of SON. The problem of resolving the Kronecker products of ordinary representations of Sn has received considerable attention, and techniques have been developed that obviate the need to use explicit character tables (Murnaghan 1937, 1938, Littlewood 1958a,b, Butler and King 1973). Furthermore many of the results have been given in an n-independent form using a "reduced notation" for labelling the irreducible representations of Sn. But the spin (or projective) irreps of Sn have received far less attention. As long ago as 1911, Issia Schur, having previously investigated the representations of any finite group by linear fractional (Schur 1904, 1907) directed his attention to the study of spin representation of Sn (Schur 1911) . Methods of constructing spin characters tables of Sn are of recent origin (Morris 1962a, Read 1977). Remarkably little is known about the resolution of Kronecker products involving the spin representations apart from the explicit use of character tables. This contrasts strongly with the corresponding reduction of the ordinary representation of Sn. In chapter 5 I give attention to the problems mentioned above. I establish an On ⥰Sn embedding and the formation of branching rules for On “ Sn is then considered, leading to a reduced notation for the spin representations of Sn, making possible many n-independent results. The first application is to discuss the n-independence of the dimensions of the spin representations of Sn. In order to facilitate the reduced notation, a special Young raising operator Roj is introduced. These results, together with consideration of difference characters of Sn, give a general procedure for resolving arbitrary Kronecker products without the explicit use of character tables. We are then able to use the method of plethysm to resolve Kronecker squares of the spin representations into their symmetric and antisymmetric parts, and eventually to classify the spin irreps as to their orthogonal, symplectic or complex characters. In the last chapter, chapter 6, I extend the reduced notation developed in chapter 5 to An, the subgroup of Sn, leading to an essentially n-independent treatment of the properties of the representations of An. Branching rules for Sn ＋ An are developed. The difference character for the irreducible representation of An are established and used to establish a series of algorithms for evaluating Kronecker products and plethysms of spin and ordinary irreps of An. In the concluding section the systematic classification of the irreps of An is given.