Olivier Dudas, Nicolas Jacon, “Alvis–Curtis Duality for Finite General Linear Groups and a Generalized Mullineux Involution”, SIGMA, 2018, Volume 14,007, 18 pp.

This article is cited in 2 papers

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

Olivier Dudas^a, Nicolas Jacon^b

^a Université Paris Diderot, UFR de Mathématiques, Bâtiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris CEDEX 13, France
^b Université de Reims Champagne-Ardenne, UFR Sciences exactes et naturelles, Laboratoire de Mathématiques EA 4535, Moulin de la Housse BP 1039, 51100 Reims, France

Abstract: 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 $\ell$.

Keywords: Mullineux involution; Alvis–Curtis duality; crystal graph; Harish-Chandra theory.

MSC: 20C20; 20C30; 05E10

Received: June 17, 2017; in final form January 22, 2018; Published online January 30, 2018

Language: English

DOI: 10.3842/SIGMA.2018.007