Computing Hecke Operators for Arithmetic Subgroups of General Linear Groups

TL;DR

This paper introduces a new algorithm for computing Hecke operators on the cohomology of arithmetic subgroups of general linear groups, applicable in all degrees and for various coefficient systems.

Contribution

The authors develop a novel algorithm that extends existing methods to compute Hecke operators on cohomology in all degrees for arithmetic subgroups of GL_n, including over number fields and division algebras.

Findings

Algorithm successfully implemented for SL_n(Z) with n=2,3,4

Results obtained for congruence subgroups of SL_3(Z)

Extends cohomology computation to all degrees using the well-tempered complex

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 $\mathrm{SL}_3(\mathbb{Z})$.