Some Properties of Generalized Foulkes Module

TL;DR

This paper introduces a new approach to analyze the decomposition of Generalized Foulkes modules into irreducible components by examining their restriction to a maximal Young subgroup, addressing an open problem in representation theory.

Contribution

It proposes a novel method for studying the decomposition of Generalized Foulkes modules via restriction to a specific Young subgroup, extending previous understanding.

Findings

Provides a new framework for decomposing Generalized Foulkes modules.

Offers insights into the structure of these modules through subgroup restriction.

Addresses an open problem in the representation theory of symmetric groups.

Abstract

Describing the decomposition of Foulkes module $F_b^a$ into irreducible Specht modules is an open problem for $a,b > 3$. In this article we provide a new approach for the Generalized Foulkes module $F_{\nu}^a$ (with arbitrary partition $\nu$ of $b$) through its restriction to a maximal Young subgroup ${S_b \times S_{ab -b}}$.