The Mullineux conjecture is about computing the p-regular partition associated with the tensor product of an irreducible representation of a symmetric group with the sign representation. Since being formulated in 1979, the conjecture attracted a lot of attention and was not settled until 1997 when B. Ford and A. Kleshchev first proved it in a paper over a hundred pages. The proof was soon been shorten and, at the same time, its quantum version was also settled. The main ingredient of the proof is the modular branching rules.
In 2003, J. Brundan and J. Kujawa discovered a proof using naturally representations of the general linear supergroup. I am going to talk about how to use the quantum linear supergroup to resolve the quantum Mullineux conjecture. This is joint work with Yanan Lin and Zhongguo Zhou.