Branching rules for modular projective representations of the symmetric groups

Summary

Abstract

In this survey we shall discuss known results concerning the decomposition numbers for the p-modular projective representations of the symmetric groups (for p odd). Several open questions are mentioned.

Introduction

It is well-known (see [4]), that the problem of determining the decomposition numbers for p-modular projective representations of the symmetric groups (for odd p) is the same problem as determining the decomposition numbers for the faithful representations of a double cover of the symmetric groups. There are, in general, two non-isomorphic such double covers, one of which is denoted by S(n) and is generated by elements t₁, …, t_n-1 together with a central involution z satisfying the relations

A result of G. D. James [6] shows that it is possible to index the irreducible p-modular Brauer characters and the ordinary characters in such a way that the decomposition matrix is upper uni-triangular. This is not the case for projective representations, and we first use the modular atlas [7] to look at some of the situations that arise. The situation is more complicated here than in the linear case since no analogue has yet been found of the construction of the irreducible modules given by Specht modules. It is a well-known result of Schur (see Theorem 8.6 of [4]) that the irreducible complex projective representations of the symmetric group S(n) may be labelled by partitions of n into distinct parts. However, this method of labelling is not bijective. If λ has an even number of even parts then there is a unique irreducible associated with λ denoted by 〈λ〉.