### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 007, 18 pages      arXiv:1706.04743      https://doi.org/10.3842/SIGMA.2018.007
Contribution to the Special Issue on the Representation Theory of the Symmetric Groups and Related Topics

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

Olivier Dudas a and 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

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

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 $\ell$.

Key words: Mullineux involution; Alvis-Curtis duality; crystal graph; Harish-Chandra theory.

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