A progenerator for representations of SL(n,q) in transverse characteristic

TL;DR

This paper proves a Morita equivalence between the group algebra of SL(n,q) and a certain idempotent subalgebra over a ring where p is invertible, enhancing understanding of their module categories in transverse characteristic.

Contribution

It establishes a Morita equivalence between RG and eRGe for SL(n,q) in transverse characteristic, providing new insights into their representation theory.

Findings

RG and eRGe are Morita equivalent.

The natural functor eM induces an equivalence of module categories.

Results apply to groups SL(n,q), GL(n,q), and PGL(n,q).

Abstract

Let G=GL(n,q), SL(n,q) or PGL(n,q) where q is a power of some prime number p, let U denote a Sylow p-subgroup of G and let R be a commutative ring in which p is invertible. Let D(U) denote the derived subgroup of U and let e be the central primitive idempotent of the group algebra RD(U) corresponding to the projection on the invariant RD(U)-submodule. The aim of this note is to prove that the R-algebras RG and eRGe are Morita equivalent (through the natural functor sending an RG-module M to the eRGe-module eM).