On the representation theory of the symmetric groups

Abstract

The purpose of this paper is to look at some results in the representation theory of the symmetric groups, both old and recent, from a modern point of view. In the first two sections we construct the irreducible representations of the symmetric groups as left ideals in the group ring. The irreducible modular representations were first constructed by James [ 131. We prove inequivalence by a method more natural than the original. The approach may be compared with that in [6]. A comprehensive survey of recent developments has been published by their originator [ 151. In Section 3 we look at the skew representations. Their dimension may be established by the same method as that used to obtain the dimenssions of the Specht modules in [lo]. A further study of skew representations may be found in [ 161. The induction algebra of symmetric groups is defined in Section 4. It is proved to be a polynomial algebra, a result also obtained in [9]. The algebra is used to give a new proof of Chung’s conjecture that the number of modular irreducible representations in a block is independent of the core. In the last two sections we examine Taulbee’s construction [7] of a class of projective, indecomposable characters. The construction uses the process of r-induction (which has been used to calculate decomposition numbers of