Cohomology of Hecke algebras

Abstract

We compute the cohomology $H^*(\mathcal{H},k)=\rm{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.