On decomposition numbers and Alvis-Curtis duality

TL;DR

This paper establishes that understanding the modular Alvis-Curtis duality for general linear groups and q-Schur algebras is equivalent to knowing their decomposition numbers over fields with characteristic not dividing q.

Contribution

It proves the equivalence between the modular Alvis-Curtis duality and decomposition numbers for these algebraic structures.

Findings

Modular Alvis-Curtis duality determines decomposition numbers.

Knowledge of duality and decomposition numbers are interchangeable.

Results apply to general linear groups and q-Schur algebras.

Abstract

We show that for general linear groups ${\rm GL}_n(q)$ as well as for $q$-Schur algebras the knowledge of the modular Alvis-Curtis duality over fields of characteristic $\ell$, $\ell \nmid q$, is equivalent to the knowledge of the decomposition numbers.