A Proof of the Mullineux Conjecture

A partition λ of a positive integer n is a sequence λ1 λ2 λm 0 of integers such that ∑λi n. For a positive integer p, a partition λ λ1 λ2 λm (or its Young diagram) is called p-regular if it does not have p or more equal parts, i.e. if there does not exist t m p 1 with λt λt 1 λt p 1. Let F be a field of characteristic p 0. It is well known that irreducible representations of the symmetric group Sn over F are naturally parametrized by pregular partitions of n (cf. for example [9, 12]). If λ is such a partition we denote the corresponding irreducible module by Dλ. Let sgnn be the one-dimensional sign representation of Sn over F; i.e., sgnn F as a vector space and g f sgn g f for any g Sn f F . Here sgn g is just the sign of the permutation g. It is clear that for any irreducible Dλ, the tensor product Dλ sgnn is also irreducible. The problem, usually called the problem of Mullineux, is to find the p-regular partition μ such that Dλ sgnn D μ. Put Dλ sgnn D bn λ In this way a bijection bn on the set Pn of p-regular partitions of n is defined for each positive integer n, and the problem is: