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# Homology, Homotopy and Applications

## Volume 12 (2010)

### Number 2

### Cohomology of Hecke algebras

Pages: 353 – 370

DOI: http://dx.doi.org/10.4310/HHA.2010.v12.n2.a12

#### Authors

#### Abstract

We compute the cohomology $H^*(\mathcal{H},k)=\mathrm{Ext}^*_{\mathcal{H}}(k,k)$ where $\mathcal{H}=\mathcal{H}(n,q)$ is the Hecke algebra of the symmetric group $\mathfrak{S}_n$ at a primitive $\ell$th root of unity $q$, and $k$ is a field of characteristic zero. The answer is particularly interesting when $\ell=2$, which is the only case where it is not graded commutative. We also carry out the corresponding computation for Hecke algebras of type $B_n$ and $D_n$ when $\ell$ is odd.

#### Keywords

Hecke algebra; cohomology ring

#### 2010 Mathematics Subject Classification

20C08

Published 1 January 2010