  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.
