The equivariant cohomology ring of the representation variety $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C}))$

TL;DR

This paper provides a detailed algebraic description of the equivariant cohomology ring of the representation variety of two commuting matrices in GL_n(C), revealing its generators, relations, and torsion-free structure.

Contribution

It offers the first explicit presentation of the equivariant cohomology ring for this variety, including generators, relations, and non-equivariant case over fields with suitable characteristic.

Findings

The ring is torsion free.

It is generated by 3n elements.

Relations are given by saturation of an n-generator ideal with Vandermonde powers.

Abstract

We give a presentation of the $\mathrm{GL}_n(\mathbb{C})$-equivariant cohomology ring with $\mathbb{Z}$-coefficients of the variety $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C})) \subseteq \mathrm{GL}_n(\mathbb{C})^2$ for any $n$. It is torsion free and minimally generated as a $H^\ast B\mathrm{GL}_n(\mathbb{C})$-algebra by $3n$ elements. The ideal of relations is the saturation of an $n$-generator ideal by even powers of the Vandermonde polynomial. For coefficients in a field whose characteristic does not divide $n!$, we also give a presentation of the non-equivariant cohomology ring of $\mathrm{Hom}(\mathbb{Z}^2,\mathrm{GL}_n(\mathbb{C}))$.