Ribbon blocks for centraliser algebras of symmetric groups

Abstract

Suppose $l,m$ are natural numbers with $l\le m$, and $\mathbb{F}$ a field of characteristic $p$, and let ${\mathcal{C}}_{l,m}^{\mathbb{F}}$ denote the centraliser of the group algebra $\mathbb{F}{S}_l$ inside $\mathbb{F}{S}_m$. Ellers and Murray give a conjectured classification of the blocks of ${\mathcal{C}}_{l,m}^{\mathbb{F}}$, in terms of the $p$-blocks of $S_l$ and $S_m$. We prove this conjecture for a family of blocks that we call ribbon blocks and belt blocks. These are the blocks containing Specht modules labelled by skew partitions having no repeated entries in their $p$-content.