Representations of symmetric groups

In this thesis we study the ordinary and the modular representation theory of the symmetric group. In particular we focus our work on different important open questions in the area. 1. Foulkes’ Conjecture In Chapter 2 we focus our attention on the long standing open problem known as Foulkes’ Conjecture. We use methods from character theory of symmetric groups to determine new information on the decomposition into irreducible characters of the Foulkes character. 2. Foulkes modules and decomposition numbers The decomposition matrix of a finite group in prime characteristic p records the multiplicities of its p-modular irreducible representations as composition factors of the reductions modulo p of its irreducible representations in characteristic zero. In Chapter 3 we give a combinatorial description of certain columns of the decomposition matrices of symmetric groups in odd prime characteristic. The result applies to blocks of arbitrarily high p-weight. It is obtained by studying the p-local structure of certain twists by the sign character of the Foulkes module H(2 n). This is joint work with Mark Wildon. In Chapter 4 we extend the results obtained in Chapter 3 on the modular structure of H(2 n), to the entire class of Foulkes modules H(a n) defined over any field F of odd prime characteristic p such that a < p 6 n. In particular we characterize the vertices of all the indecomposable summands of H(a n). 3. Vertices of Specht and simple modules In Chapter 5 we study the vertices of indecomposable Specht modules for symmetric groups. For any given indecomposable non-projective Specht module, the main theorem of the chapter describes a large p-subgroup contained in its vertex. In Chapter 6 we consider the vertices of simple modules for the symmetric groups in prime characteristic p. The main theorem of the chapter completes the classification of the vertices of simple FSn-modules labelled by hook partitions. This is joint work with Susanne Danz.