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Hilbert-Poincare series for spaces of commuting elements in Lie groups

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 Added by Mentor Stafa
 Publication date 2017
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and research's language is English




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In this article we study the homology of spaces ${rm Hom}(mathbb{Z}^n,G)$ of ordered pairwise commuting $n$-tuples in a Lie group $G$. We give an explicit formula for the Poincare series of these spaces in terms of invariants of the Weyl group of $G$. By work of Bergeron and Silberman, our results also apply to ${rm Hom}(F_n/Gamma_n^m,G)$, where the subgroups $Gamma_n^m$ are the terms in the descending central series of the free group $F_n$. Finally, we show that there is a stable equivalence between the space ${rm Comm}(G)$ studied by Cohen-Stafa and its nilpotent analogues.



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In this paper we study homological stability for spaces ${rm Hom}(mathbb{Z}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $ngeqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, ${rm Comm}(G)$ and ${rm B_{com}} G$, introduced by Cohen-Stafa and Adem-Cohen-Torres-Giese respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability - in particular, the theory of ${rm FI}_W$-modules developed by Church-Ellenberg-Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.
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