Mullineux involution and crystal isomorphisms

We develop a new approach for the computation of Mullineux involution symmetric group and its Hecke algebra using notion crystal isomorphism Iwahori-Matsumoto affine type \(A\). As consequence, we obtain several elementary combinatorial algorithms computation, one which is equivalent to Xu algorithm (and thus original algorithm). simple interpretation these proof that they indeed compute involution.Mathematics Subject Classifications: 20C08, 05E10Keywords: Symmetric group, involution, graph