Crystals, regularisation and the Mullineux map

TL;DR

This paper introduces a new combinatorial description of the Mullineux map using crystal isomorphisms and regularisation maps, providing simpler proofs of existing conjectures and theorems in representation theory.

Contribution

It offers a novel combinatorial framework for the Mullineux map via crystal isomorphisms and generalised regularisation maps, enhancing understanding and proofs in modular representation theory.

Findings

New combinatorial description of the Mullineux map

Purely combinatorial proofs of Lyle's conjecture and Paget's theorem

Connections established between crystal isomorphisms and regularisation maps

Abstract

The Mullineux map is a combinatorial function on partitions which describes the effect of tensoring a simple module for the symmetric group in characteristic $p$ with the one-dimensional sign representation. It can also be interpreted as an isomorphism between crystal graphs for $\widehat{\mathfrak{sl}}_p$. We give a new combinatorial description of the Mullineux map by expressing this crystal isomorphism as a composition of isomorphisms between different crystals. These isomorphisms are defined in terms of new generalised regularisation maps introduced by Millan Berdasco. We then given two applications of our new realisation of the Mullineux map, by providing purely combinatorial proofs of a conjecture of Lyle relating the Mullineux map with regularisation, and a theorem of Paget describing the Mullineux map in RoCK blocks of symmetric groups.