Characters of odd degree of symmetric groups

We construct a natural bijection between odd-degree irreducible characters of S_n and linear characters of its Sylow 2-subgroup P_n. When n is a power of 2, we show that such a bijection is nicely induced by the restriction functor. We conclude with a characterization of the irreducible characters \chi of S_n such that the restriction of \chi to P_n has a unique linear constituent

Vertices of simple modules of symmetric groups labelled by hook partitions

In this article we study the vertices of simple modules for the symmetric groups in prime characteristic $p$. In particular, we complete the classification of the vertices of simple $S_n$-modules labelled by hook partitions

Foulkes modules and decomposition numbers of the symmetric group

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. The main theorem of this paper gives 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 weight. It is obtained by studying the p-local structure of certain twists of the permutation module given by the action of the symmetric group of even degree 2m on the collection of set partitions of a set of size 2m into m sets each of size two. In particular, the vertices of the indecomposable summands of all such modules are characterized; these summands form a new family of indecomposable p-permutation modules for the symmetric group. As a further corollary it is shown that for every natural number w there is a diagonal Cartan number in a block of the symmetric group of weight w equal to w+1

A note on restriction of Characters of Alternating groups to Sylow Subgroups

We restrict irreducible characters of alternating groups of degree divisible by $p$ to their Sylow $p$-subgroups and study the number of linear constituents

On the decomposition of the Foulkes module

The Foulkes module H^(a^b) is the permutation module for the symmetric group S_ab given by the action of S_ab on the collection of set partitions of a set of size ab into b sets each of size a. The main result of this paper is a sufficient condition for a simple CS_{ab}-module to have zero multiplicity in H^(a^b). A special case of this result implies that no Specht module labelled by a hook partition (ab - r, 1^r) appears in H(a^b)