Mullineux involution and crystal isomorphisms

TL;DR

This paper introduces a novel approach to compute the Mullineux involution using crystal isomorphisms and affine Hecke algebra involutions, leading to new combinatorial algorithms and simpler proofs.

Contribution

It develops a new method for Mullineux involution computation based on crystal isomorphisms, providing elementary algorithms and proofs.

Findings

New combinatorial algorithms for Mullineux involution

Equivalence of one algorithm to Xu's algorithm

Elementary proof of algorithm correctness

Abstract

We develop a new approach for the computation of the Mullineux involution for the symmetric group and its Hecke algebra using the notion of crystal isomorphism and the Iwahori-Matsumoto involution for the affine Hecke algebra of type A. As a consequence, we obtain several new elementary combinatorial algorithms for its computation, one of which is equivalent to Xu's algorithm (and thus Mullineux' original algorithm). We thus obtain a simple interpretation of these algorithms and a new elementary proof that they indeed compute the Mullineux involution.