Support varieties for Hecke algebras

TL;DR

This paper develops a support variety theory for Hecke algebras of symmetric groups, linking homological properties to geometric objects, with computational methods and applications to various modules.

Contribution

It introduces a canonical support variety framework for Hecke algebras, enabling tractable computations and extending to other classical groups.

Findings

Support varieties detect module complexity.

Explicit calculations for permutation, Young, and Specht modules.

Extension potential to other classical Hecke algebras.

Abstract

Let ${\mathcal H}_{q}(d)$ be the Iwahori-Hecke algebra for the symmetric group, where $q$ is a primitive $l$th root of unity. In this paper we develop a theory of support varieties which detects natural homological properties such as the complexity of modules. The theory the authors develop has a canonical description in an affine space where computations are tractable. The ideas involve the interplay with the computation of the cohomology ring due to Benson, Erdmann and Mikaelian, the theory of vertices due to Dipper and Du, and branching results for cohomology by Hemmer and Nakano. Calculations of support varieties and vertices are presented for permutation, Young and classes of Specht modules. Furthermore, a discussion of how the authors' results can be extended to other Hecke algebras for other classical groups is presented at the end of the paper.