栅栏偏序集、好分次与Frobenius极大抛物子代数

中文摘要 AI 辅助

设 $\mathfrak L$ 是 $\mathfrak{sl}_n$ 的一个Frobenius极大抛物子代数。对于任意 $F\in\mathfrak L^*$ 使得Kirillov形式 $B_F(x,y)=F([x,y])$ 非退化，令 $\widehat F$ 表示相应的主元素。我们证明 $\operatorname{ad}_{\widehat F}$ 在 $\mathfrak L$ 上的特征值重数构成一个关于 $\frac12$ 对称的单峰序列。我们还证明 $\operatorname{ad}_{\widehat F}$ 在 $\mathfrak{gl}_n$ 上的特征值重数构成一个关于 $0$ 对称的单峰序列。证明通过Panyushev约化将 $\mathfrak L$ 关联的秩化蛇形与一个相关栅栏偏序集的序理想联系起来。栅栏偏序集的秩多项式的已知单峰性蕴含了蛇形的单峰性，进而决定了 $\mathfrak{gl}_n$ 的一个在Elashvili和Kac意义下的好分次。我们证明该分次与由主元素诱导的分次一致，并且与该分次关联的金字塔可以被填充，使得其好元素 $e$ 位于 $\mathfrak L$ 中。两个单峰性结果随后由好分次带来的 $\operatorname{ad}_e$ 的单射性质以及由双线性形式 $B_F$ 诱导的对偶性得出。

英文摘要

Let $\mathfrak L$ be a Frobenius maximal parabolic subalgebra of $\mathfrak{sl}_n$. For any $F\in\mathfrak L^*$ for which the Kirillov form $B_F(x,y)=F([x,y])$ is non-degenerate, let $\widehat F$ denote the associated principal element. We prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat F}$ on $\mathfrak L$ form a unimodal sequence symmetric about $\frac12$. We also prove that the multiplicities of the eigenvalues of $\operatorname{ad}_{\widehat F}$ on $\mathfrak{gl}_n$ form a unimodal sequence symmetric about $0$. The proof relates the ranked meander associated to $\mathfrak L$ to the order ideals of a related fence poset through Panyushev reduction. The known unimodality of the rank polynomial of the fence poset implies that of the meander, which in turn determines a good grading of $\mathfrak{gl}_n$ in the sense of Elashvili and Kac. We prove that this grading coincides with that induced by the principal element and that the pyramid associated to this grading may be filled in such a way that its good element $e$ lies in $\mathfrak L$. The two unimodality results then follow from the injectivity properties of $\operatorname{ad}_e$ coming from the good grading and the duality induced by the bilinear form $B_F$.

发表机构

• Loyola University Chicago（芝加哥洛约拉大学）
• Saint Mary’s University（圣玛丽大学）

机构由 AI 辅助整理，请以论文原文为准。