Homological stability for Iwahori-Hecke algebras

TL;DR

This paper proves homological stability for Iwahori-Hecke algebras of type A, extending known results for symmetric groups and pioneering the application of homological stability techniques to non-group algebras.

Contribution

It establishes homological stability for Iwahori-Hecke algebras, a novel application of these techniques beyond traditional group algebra contexts.

Findings

Homological stability holds for H_n of type A_{n-1}.

Recovers Nakaoka's stability for symmetric groups at parameter 1.

First application of homological stability techniques to non-group algebras.

Abstract

We show that the Iwahori-Hecke algebras H_n of type A_{n-1} satisfy homological stability, where homology is interpreted as an appropriate Tor group. Our result precisely recovers Nakaoka's homological stability result for the symmetric groups in the case that the defining parameter is equal to 1. We believe that this paper, and our joint work with Boyd on Temperley-Lieb algebras, are the first time that the techniques of homological stability have been applied to algebras that are not group algebras.