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

令G为经典群，令$\mathrm{坎}({\mathbb{Z}}^{米},G)$表示在G中交换m元组的空间。首先，我们改进 Poincaré 级数的公式$\mathrm{坎}({\mathbb{Z}}^{米},G)$由于 Ramras 和 Stafa 通过将（有符号）整数分区分配给（有符号）排列。使用改进后的公式，我们确定了 Poincaré 级数的顶项，并应用它来证明$\mathrm{坎}({\mathbb{Z}}^{米},G)$关于m的奇偶性和有理双曲线$\mathrm{坎}({\mathbb{Z}}^{米},G)$ 为了 $米\ge 2$. 接下来，我们给出了上同调的最小生成集$\mathrm{坎}({\mathbb{Z}}^{米},G)$并确定低维的上同调。我们应用这些结果来证明同源稳定性$\mathrm{坎}({\mathbb{Z}}^{米},G)$具有最佳的稳定范围。贝尔德证明了$\mathrm{坎}({\mathbb{Z}}^{米},G)$被识别为G的 Weyl 群的某个不变量环，我们的方法是直接计算这个不变量环。