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On the structure of Foulkes modules for the symmetric group

de Boeck, Melanie (2015) On the structure of Foulkes modules for the symmetric group. Doctor of Philosophy (PhD) thesis, University of Kent.

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Abstract

This thesis concerns the structure of Foulkes modules for the symmetric group. We study `ordinary' Foulkes modules $H^{(m^n)}$, where $m$ and $n$ are natural numbers, which are permutation modules arising from the action on cosets of $\mathfrak{S}_m\wr\mathfrak{S}_n\leq \mathfrak{S}_{mn}$. We also study a generalisation of these modules $H^{(m^n)}_\nu$, labelled by a partition $\nu$ of $n$, which we call generalised Foulkes modules. Working over a field of characteristic zero, we investigate the module structure using semistandard homomorphisms. We identify several new relationships between irreducible constituents of $H^{(m^n)}$ and $H^{(m^{n+q})}$, where $q$ is a natural number, and also apply the theory to twisted Foulkes modules, which are labelled by $\nu=(1^n)$, obtaining analogous results. We make extensive use of character-theoretic techniques to study $\varphi^{(m^n)}_\nu$, the ordinary character afforded by the Foulkes module $H^{(m^n)}_\nu$, and we draw conclusions about near-minimal constituents of $\varphi^{(m^n)}_{(n)}$ in the case where $m$ is even. Further, we prove a recursive formula for computing character multiplicities of any generalised Foulkes character $\varphi^{(m^n)}_\nu$, and we decompose completely the character $\varphi^{(2^n)}_\nu$ in the cases where $\nu$ has either two rows or two columns, or is a hook partition. Finally, we examine the structure of twisted Foulkes modules in the modular setting. In particular, we answer questions about the structure of $H^{(2^n)}_{(1^n)}$ over fields of prime characteristic.

Item Type: Thesis (Doctor of Philosophy (PhD))
Thesis advisor: Paget, Rowena E.
Uncontrolled keywords: Symmetric group Foulkes module Characters Plethysm
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Users 1 not found.
Date Deposited: 18 Aug 2015 15:00 UTC
Last Modified: 29 May 2019 15:56 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/50212 (The current URI for this page, for reference purposes)
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