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

David Benson (Institute of Mathematics, University of Aberdeen, Scotland, United Kingdom)

Karin Erdmann (Mathematical Institute, University of Oxford, United Kingdom)

Aram Mikaelian (Mathematical Institute, University of Oxford, United Kingdom)

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

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