The simplicial complex of Brauer pairs of a finite reductive group

TL;DR

This paper investigates the topological structure of the simplicial complex formed by Brauer pairs in finite reductive groups, revealing its homotopy equivalence to known complexes in different characteristic cases.

Contribution

It extends previous work by showing the homotopy type of the Brauer pair complex aligns with the Tits building in defining characteristic and with a generalized Harish-Chandra complex in non-defining characteristic.

Findings

Homotopy type matches Tits building in defining characteristic

Homotopy equivalent to a generalized Harish-Chandra complex in non-defining characteristic

Extends earlier results on the Brown complex using connected subpairs and twisted block induction

Abstract

In this paper we study the simplicial complex induced by the poset of Brauer pairs ordered by inclusion for the family of finite reductive groups. In the defining characteristic case, the homotopy type of this simplicial complex coincides with that of the Tits building thanks to a well-known result of Quillen. On the other hand, in the non-defining characteristic case, we show that the simplicial complex of Brauer pairs is homotopy equivalent to a simplicial complex determined by generalised Harish-Chandra theory. This extends earlier results of the author on the Brown complex and makes use of the theory of connected subpairs and twisted block induction developed by Cabanes and Enguehard.