Spaces of commuting elements in the classical groups

TL;DR

This paper refines the understanding of the topology and cohomology of spaces of commuting elements in classical groups, revealing their dependence on parameters and establishing homological stability.

Contribution

It provides a refined formula for the Poincaré series, determines the cohomology structure, and proves homological stability for these spaces, advancing the algebraic and topological understanding.

Findings

Refined Poincaré series formula for Hom$(\

,

,

Abstract

Let $G$ be the classical group, and let Hom$(\mathbb{Z}^m,G)$ denote the space of commuting $m$-tuples in $G$. First, we refine the formula for the Poincar\'e series of Hom$(\mathbb{Z}^m,G)$ due to Ramras and Stafa by assigning (signed) integer partitions to (signed) permutations. Using the refined formula, we determine the top term of the Poincar\'e series, and apply it to prove the dependence of the topology of Hom$(\mathbb{Z}^m,G)$ on the parity of $m$ and the rational hyperbolicity of Hom$(\mathbb{Z}^m,G)$ for $m\ge 2$. Next, we give a minimal generating set of the cohomology of Hom$(\mathbb{Z}^m,G)$ and determine the cohomology in low dimensions. We apply these results to prove homological stability for Hom$(\mathbb{Z}^m,G)$ with the best possible stable range. Baird proved that the cohomology of Hom$(\mathbb{Z}^m,G)$ is identified with a certain ring of invariants of the Weyl group of $G$, and our approach is a direct calculation of this ring of invariants.