Crystals, regularisation and the Mullineux map

The Mullineux map is a combinatorial function on partitions which describes the effect of tensoring simple module for symmetric group in characteristic $p$ with one-dimensional sign representation. It can also be interpreted as signed isomorphism between crystal graphs $\widehat{\mathfrak{sl}}\_p$. We give new description by expressing this composition isomorphisms different crystals. These are defined terms generalised regularisation maps introduced Millan Berdasco. then given two applications our realisation map, providing purely proofs conjecture Lyle relating regularisation, and theorem Paget describing RoCK blocks groups.