The asymptotics of tensor powers of Foulkes modules

Abstract

In this paper we study the modular structure of the Foulkes module $H^{(2^m)}$ of the symmetric group $S_{2m}$ acting on set partitions of a set of size 2m into m sets each of size 2, defined over a field of positive characteristic p. We aim to understand the decomposition of the tensor powers of $H^{(2^m)}$. In particular, we study the asymptotics of the non-projective part of tensor powers of these modules by computing the gamma invariant as defined by Dave Benson and Peter Symonds.

Keywords:

• Foulkes module
• permutation module
• symmetric group
• tensor product

2020 Mathematics Subject Classification:

• Primary: 20C30
• 20C20
• Secondary: 05E10