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 $n\geqslant 1$, 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.

中文翻译：

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李群交换元空间的同调稳定性

在本文中，我们研究了李群 $G$ 中成对交换 $n$-元组的空间 ${\rm Hom}(\mathbb{Z}^n,G)$ 的同调稳定性。我们证明，对于每个 $n\geqslant 1$，这些空间满足有理同调稳定性，因为 $G$ 范围通过紧致、连通李群或其复化的任何经典序列。我们证明了有理等变同调、字符变体以及这些空间的无限维类似物的类似结果，${\rm Comm}(G)$ 和 ${\rm B_{com}} G$，由 Cohen 引入-分别是 Stafa 和 Adem-Cohen-Torres-Giese。此外，我们表明固定群$G$ 中无序通勤$n$-元组空间的有理同调随着$n$ 的增加而稳定。我们的证明使用了表征稳定性理论——特别是，由 Church-Ellenberg-Farb 和 Wilson 开发的 ${\rm FI}_W$ 模块理论。在所有这些结果中，我们获得了稳定范围的特定界限，并且我们证明了同构同构是由空间映射引起的。