On the structure of Foulkes modules for the symmetric group

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.