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

## TL;DR

This paper explores how Alvis-Curtis duality affects unipotent representations of finite general linear groups, revealing a connection to a generalized Mullineux involution that depends on the prime and the field size.

## Contribution

It introduces a generalized Mullineux involution that describes the permutation of simple modules induced by Alvis-Curtis duality for $ ext{GL}_n(q)$ in non-defining characteristic.

## Key findings

- Permutation of simple modules expressed via the generalized Mullineux involution.
- The involution depends on the prime $ ext{ell}$ and the order of $q$ modulo $ ext{ell}$.
- Provides a new combinatorial tool for understanding representations of finite groups of Lie type.

## 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 $\ell$.