m at h . G R / 0 21 01 08 v 1 7 O ct 2 00 2 A NEW PROOF OF THE MULLINEUX CONJECTURE

Let Sn be the symmetric group on n letters, k be a field of characteristic p and D be the irreducible kSn-module corresponding to a p-regular partition λ of n, as in [12]. By tensoring D with the 1-dimensional sign representation we obtain another irreducible kSn-module. If p = 0, D λ ⊗ sgn ∼= D ′ , where λ′ is the conjugate of the partition λ, and if p = 2, we obviously have that D ⊗ sgn ∼= D. In all other cases, it is surprisingly difficult to describe the partition labeling the irreducible module D ⊗ sgn combinatorially. In 1979, Mullineux [22] gave an algorithmic construction of a bijection M on p-regular partitions, and conjectured that D ⊗ sgn ∼= DM(λ). Mullineux’s conjecture was finally proved in 1996. The key breakthrough leading to the proof was made in [16], when Kleshchev discovered an alternative algorithm, quite different in nature to Mullineux’s, and proved that it computes the label of D ⊗ sgn. Then Ford and Kleshchev [10] proved combinatorially that Kleshchev’s algorithm was equivalent to Mullineux’s, hence proving the Mullineux conjecture. Since then, different and easier approaches to the combinatorial part of the proof, i.e. that Kleshchev’s algorithm equals Mullineux’s algorithm, have been found by Bessenrodt and Olsson [3] and by Xu [29]. Also Lascoux, Leclerc and Thibon [18] have used Ariki’s theorem [1] to give a different proof of the results of [16]. The purpose of the present article is to explain a completely different proof of the Mullineux conjecture. In [28], Xu discovered yet another algorithm, and gave a short combinatorial argument to show that it was equivalent to Mullineux’s original algorithm. We will show directly from representation theory that Xu’s algorithm computes the label of D⊗ sgn. In this way, we obtain a relatively direct proof of the Mullineux conjecture that bypasses Kleshchev’s algorithm altogether. The idea behind our approach is a simple one. There is a superalgebra analogue of Schur-Weyl duality relating representations of Sn to representations of the supergroup GL(n|n). Moreover, there is an involution on representations of GL(n|n) induced by twisting with its natural outer automorphism, which corresponds under Schur-Weyl duality to tensoring with the sign representation. Ideas of Serganova [25] give an easy-to-prove algorithm for computing this involution, hence by Schur-Weyl duality we obtain an algorithm for computing