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# Homology, Homotopy and Applications

## Volume 21 (2019)

### Number 2

### Support varieties for Hecke algebras

Pages: 59 – 82

DOI: http://dx.doi.org/10.4310/HHA.2019.v21.n2.a5

#### Authors

#### Abstract

Let $\mathcal{H}_q (d)$ be the Iwahori–Hecke algebra for the symmetric group, $\Sigma_d$, 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.

#### Keywords

support variety, cohomology, Hecke algebra

#### 2010 Mathematics Subject Classification

20C30

Research of the first author was supported in part by NSF grant DMS-1701768. Research of the second author was supported in part by NSF RTG grant DMS-1344994.

Received 2 February 2018

Received revised 22 August 2018

Published 19 December 2018