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Cohomology of Hecke algebras. (English) Zbl 1236.20002

Summary: We compute the cohomology \(H^*(\mathcal H,k)=\text{Ext}^*_{\mathcal H}(k,k)\) 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.

MSC:

20C08 Hecke algebras and their representations
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
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