Geodesic
A new proof of the Mullineux conjecture.
Journal of Algebraic Combinatorics, Tome 18 (2003) no. 1, pp. 13-39.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $S _{d}$ denote the symmetric group on $d$ letters. In 1979 Mullineux conjectured a combinatorial algorithm for calculating the effect of tensoring with an irreducible $S _{d}$-module with the one dimensional sign module when the ground field has positive characteristic. Kleshchev proved the Mullineux conjecture in 1996. In the present article we provide a new proof of the Mullineux conjecture which is entirely independent of Kleshchev's approach. Applying the representation theory of the supergroup $GL( m | n)$ and the supergroup analogue of Schur-Weyl Duality it becomes straightforward to calculate the combinatorial effect of tensoring with the sign representation and, hence, to verify Mullineux's conjecture. Similar techniques also allow us to classify the irreducible polynomial representations of $GL( m | n)$ of degree $d$ for arbitrary m, n, and $d$.
Keywords: symmetric group, Mullineux, modular representation theory, supergroups, $GL( m | n)$ 
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     title = {A new proof of the {Mullineux} conjecture.},
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Brundan, Jonathan; Kujawa, Jonathan. A new proof of the Mullineux conjecture.. Journal of Algebraic Combinatorics, Tome 18 (2003) no. 1, pp. 13-39. http://geodesic.mathdoc.fr/item/JAC_2003__18_1_a3/