Ribbon blocks for centraliser algebras of symmetric groups

TL;DR

This work advances the block theory of centraliser algebras $\mathbb{F}S_m^{S_l}$ in positive characteristic by validating Ellers–Murray's conjecture for two substantial families: ribbon blocks and belt blocks. Leveraging the bridge to the degenerate affine Hecke algebra, it constructs formal and central characters for skew Specht modules and defines $p$-shapes to track composition factors, ultimately showing that the relevant decomposition matrices are connected. Through a detailed combinatorial analysis of rim-hooks, abacus configurations, and arrow-graph refinements, the authors prove that these blocks are indeed blocks of the centraliser algebra. The results deepen our understanding of how modular block structure in centraliser algebras mirrors the $p$-block structure of symmetric groups, with implications for local-global representation theory and related conjectures.

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.

Theorems & Definitions (9)

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