Alvis-Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution

[摘要] We study the effect of Alvis-Curtis duality on the unipotent representations of $\mathrm{GL}_n(q)$ in non-defining characteristic $\ell$. We show that the permutation induced on the simple modules can be expressed in terms of a generalization of the Mullineux involution on the set of all partitions, which involves both $\ell$ and the order of $q$ modulo $\ell$.

[关键词] Mullineux involution;Alvis–Curtis duality;crystal graph;Harish-Chandra theory