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arXiv · 2010.06036

Computing Hecke Operators for Arithmetic Subgroups of General Linear Groups

Mark McConnell·Robert MacPherson
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Abstract

We present an algorithm to compute the Hecke operators on the equivariant cohomology of an arithmetic subgroup $\Gamma$ of the general linear group $\mathrm{GL}_n$. This includes $\mathrm{GL}_n$ over a number field or a finite-dimensional division algebra. As coefficients, we may use any finite-dimensional local coefficient system. Unlike earlier methods, the algorithm works for the cohomology $H^i$ in all degrees $i$. It starts from the well-rounded retract $\tilde{W}$, a $\Gamma$-invariant cell complex which computes the cohomology. It extends $\tilde{W}$ to a new well-tempered complex $\tilde{W}^+$ of one higher real dimension, using a real parameter called the temperament. The algorithm has been coded up for $\mathrm{SL}_n(\mathbb{Z})$ for $n=2,3,4$; we present some results for congruence subgroups of $\mathrm{SL}_3(\mathbb{Z})$.

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Mark McConnell, Robert MacPherson. 2020-10-12. Computing Hecke Operators for Arithmetic Subgroups of General Linear Groups. https://arxiv.org/abs/2010.06036

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