论文标题
两个轨道的结合轨道谐波
Orbit harmonics for the union of two orbits
论文作者
论文摘要
Garsia和Procesi在研究Springer的代表性时，证明了Springer纤维的共同体学环与单点$ s_n $ Orbit的坐标环的相关级环是$ \ MATHBB {C C}^n $的相关级环。这种构造是他们对弹簧表示形式分析的必不可少的工具，最近在其他几种组合和几何环境中以轨道谐波的名义重新出现。在本文中，我们分析了两个$ s_n $ orbits的结合的轨道谐波。我们证明，当两个轨道的坐标总和不同时，相应的分级$ s_n $表示是两个Springer表示的直接总和，其中一个是1个springer表示。
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.
