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

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 $\mathrm{Hom}({\mathbb{Z}}^2,{\mathrm{GL}}_n(\mathbb{C}))$.

29 pages