Foulkes modules and decomposition numbers of the symmetric group

TL;DR

This paper provides a combinatorial description of parts of the decomposition matrices of symmetric groups in odd prime characteristic, revealing new indecomposable modules and specific Cartan numbers.

Contribution

It introduces a combinatorial approach to understanding decomposition matrices and constructs a new family of indecomposable p-permutation modules for symmetric groups.

Findings

Characterization of vertices of indecomposable summands

Identification of a new family of indecomposable p-permutation modules

Existence of diagonal Cartan numbers equal to weight plus one

Abstract

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.