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 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.

Mathematics Subject Classifications: 20C08, 05E10

Keywords: Symmetric group, Mullineux involution, crystal graph