Regularisation and the Mullineux map

TL;DR

This paper classifies conjugate partition pairs whose regularisations are related by the Mullineux map, confirming a conjecture and answering a question in the representation theory of symmetric groups.

Contribution

It provides a complete classification of such partition pairs, proving a conjecture and advancing understanding of the Mullineux map's combinatorial properties.

Findings

Classification of conjugate partition pairs under the Mullineux map

Proof of Lyle's conjecture

Answer to Bessenrodt, Olsson, and Xu's question

Abstract

We classify the pairs of conjugate partitions whose regularisations are images of each other under the Mullineux map. This classification proves a conjecture of Lyle, answering a question of Bessenrodt, Olsson and Xu.