旗簇的泊松退化轨迹
The Poisson degeneracy locus of a flag variety
AI总结：
本研究针对配备标准泊松结构的复半单李群完全旗簇的泊松退化轨迹展开系统分析，给出组合描述与结构刻画，证明其不可约分支同构于复射影直线的幂次，并将结果推广至部分旗簇，明确了最高维数的组合与代数表示论意义。
摘要：
我们对所有配备标准泊松结构的复半单李群的完全旗簇的退化轨迹开展了全面研究。研究表明，约化的泊松退化轨迹在典范极大环面的作用下可分层为开Richardson簇$\u27e8\tilde{\u2113}_v^w$，对应满足$v \u2264 w$的外尔群元素对，这类元素对拓展了布吕赫序的覆盖关系。我们给出了这类元素对的四种不同组合描述，并证明它们的布吕赫区间是幂集。对应的闭Richardson簇$\u203e{\u27e8\tilde{\u2113}_v^m}$被证明与$(\u2102\u2119^1)^d$（$d \u2265 0$）同构，且与分层结构相容。由此我们得到，所有完全旗簇的约化泊松退化轨迹都是连通的，其所有不可约分支都同构于$(\u2102\u2119^1)^n$（$n$为某个非负整数）；它们一般不是等维的。借助投影Richardson簇框架，这些结果被推广到所有部分旗簇。我们证明，完全旗情形下约化泊松退化轨迹的不可约分支的最高维数等于Kostant根级联的基数与最长外尔群元素的反射长度。研究还表明，旗簇的泊松退化轨迹一般不是约化的。
英文摘要：
We present a comprehensive study of the degeneracy loci of the full flag varieties of all complex semisimple Lie groups equipped with the standard Poisson structures. The reduced Poisson degeneracy loci are shown to stratify under the action of the canonical maximal torus into open Richardson varieties $\mathcal{R}_v^w$ for pairs of Weyl group elements $v \leq w$ that extend the covering relation of the Bruhat order. Four different combinatorial descriptions of those pairs are given, and it is shown that their Bruhat intervals are power sets. The corresponding closed Richardson varieties $\overline{\mathcal{R}_v^m}$ are shown to be isomorphic to $(\mathbb{C}\mathbb{P}^1)^d$ for $d \geq 0$ in a compatible way with the stratification. As a consequence, we obtain that the reduced Poisson degeneracy loci of all full flag varieties are connected, and all of their irreducible components are isomorphic to $(\mathbb{C}\mathbb{P}^1)^n$ for some $n \geq 0$; they are not equidimensional in general. Using the framework of projected Richardson varieties, these results are extended to all partial flag varieties. The top dimension of irreducible components of the reduced Poisson degeneracy locus in the full flag case is proved to be equal to the cardinality of Kostant's cascade of roots and the reflective length of the longest Weyl group element. It is shown that the Poisson degeneracy loci of flag varieties are not reduced in general.
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