On decomposition numbers and Alvis-Curtis duality

Abstract

We show that for general linear groups ${\mathrm{GL}}_n(q)$ 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.

to appear in Math. Proc. Cam. Philos. Soc