A new proof of the Mullineux conjecture

Abstract

Let S_d be the symmetric group on d letters and let k be a field of characteristic p>2. Tensoring an irreducible S_d module with the sign representation defines an involution on the p-regular partitions of d. It is suprisingly difficult to describe this involution combinatorially. Mullineux conjectured an algorithmic description in 1979. Kleshchev gave an entirely new algorithm describing the involution in 1996 and proved with Ford that it agrees with Mullineux's. Using the modular representation theory of the supergroup GL(m|n) we provide the first new proof of the Mullineux conjecture which is independent of Kleshchev's approach. Similar techniques allow us to classify the irreducible polynomial representations of GL(m|n), completing recent partial results by Donkin.