# Cohomology of the space of commuting elements in the classical groups

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Daisuke Kishimoto, Kyoto University

Let $G$ be a compact connected Lie group. The space $\mathrm{Hom}(\mathbb{Z}^m,G)$ is defined as a subspace of $G^m$ consisting of pairwise commuting $m$-tuples. This space has diverse connections to geometry, topology and physics, and its cohomology has a deep connection to representations of the Weyl group of $G$. I will talk about a refinement of the formula for the Poincar\’e series of  $\mathrm{Hom}(\mathbb{Z}^m,G)$ given by Ramras and Stafa, when $G$ is the classical group, and give its applications. I will also talk about minimal generating set of the cohomology of  $\mathrm{Hom}(\mathbb{Z}^m,G)$ for $G$ classical, and its application to homological stability.