两个轨道并的轨道调和函数
Orbit harmonics for the union of two orbits
摘要：
Garsia和Procesi在研究Springer表示时证明了Springer纤维的上同调环同构于$\mathbb{C}^n$中一点的$S_n$轨道的坐标环的相关分次环。这种构造是他们分析施普林格表象的一个重要工具，它的变体最近在其他几个组合和几何背景下以轨道谐波的名义重新出现。在这篇文章中，我们分析了两个$S_n$轨道并的轨道谐波。我们证明了当两个轨道的坐标和不同时，对应的分次$S_n$表示是两个Springer表示的直和，其中一个表示的度移1.
英文摘要：
Garsia and Procesi, in their study of Springer's representation, proved that the cohomology ring of a Springer fiber is isomorphic to the associated graded ring of the coordinate ring of the $S_n$ orbit of a single point in $\mathbb{C}^n$. This construction was an essential tool in their analysis of the Springer representation, and variations of it have reappeared recently in several other combinatorial and geometric contexts under the name orbit harmonics. In this article, we analyze the orbit harmonics of a union of two $S_n$ orbits. We prove that when the coordinate sums of the two orbits are different, the corresponding graded $S_n$ representation is a direct sum of two Springer representations, one of which is shifted in degree by 1.
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