Ribbon blocks for centraliser algebras of symmetric groups

TL;DR

This paper proves a conjecture classifying certain blocks of centraliser algebras of symmetric groups, specifically ribbon and belt blocks, using properties of Specht modules and skew partitions.

Contribution

It establishes the classification of ribbon and belt blocks of centraliser algebras of symmetric groups, confirming a conjecture for these specific block families.

Findings

Classification of ribbon blocks proved

Classification of belt blocks proved

Conjecture confirmed for these block families

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.