论文标题
古典群体中通勤元素的空间
Spaces of commuting elements in the classical groups
论文作者
论文摘要
令$ g $为古典群体，然后让hom $（\ mathbb {z}^m，g）$表示$ g $中通勤$ m $ tuples的空间。首先，我们通过将（签名）整数分区分配给（签名）置换量来完善HOM $（\ Mathbb {Z}^M，G）的Poincaré系列的公式。使用精致的公式，我们确定了Poincaré系列的顶级，并将其应用于证明HOM $（\ Mathbb {Z}^M，G）$ $ M $的拓扑的依赖性，而HOM $（\ MATHBB {Z}^M，G）$ M $的理性超质性对于$ M $ m \ ge ge 2 $。接下来，我们给出最小的HOM $（\ Mathbb {Z}^M，G）$的共同生成集，并在低维度中确定共同体。我们将这些结果应用于HOM $（\ MATHBB {Z}^M，G）$具有最佳稳定范围的同源稳定性。 Baird证明了HOM $（\ MATHBB {Z}^M，G）$的共同体，并用$ G $的Weyl of of $ g $的某些不变性戒指确定，我们的方法是对这种不变性的直接计算。
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 Poincaré 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 Poincaré 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.
