具有 Hessenberg 函数\(h(i)\leq i + 1\)的 A 型幂零 Hessenberg 簇
Type A Nilpotent Hessenberg varieties with Hessenberg function $h(i)\le i+1$
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中文总结 AI 辅助
研究具有\(h(i)\leq i + 1\)的 A 型幂零 Hessenberg 簇即广义抛物 Peterson 簇，证明其可分解，给出最大维数分量基数与维数相关结论及 Poincaré 多项式递归公式，部分回答 Hessenberg 簇几何问题并引发新思考。
中文摘要 AI 辅助
我们研究与满足\(h(i)\leq i + 1\)的 Hessenberg 函数相关的 A 型幂零 Hessenberg 簇，称其为广义抛物 Peterson 簇。我们证明此类簇可分解为特定广义抛物 Peterson 簇\(\operatorname{Pet}_{\lambda,\alpha}\)的并集，其中\(\lambda\)是整数划分，\(\alpha\)是整数合成且\(\alpha\)被\(\lambda\)控制。当\(\alpha\)被\(\lambda\)控制时，\(\operatorname{Pet}_{\lambda,\alpha}\)的最大维数分量的基数等于 Kostka 数\(\mathcal{K}_{\lambda\alpha}\)，其维数仅由\(\lambda\)和\(\alpha\)的长度确定。我们给出了\(\operatorname{Pet}_{\lambda,\alpha}\)的 Poincaré 多项式的递归公式。这些结果部分回答了关于 Hessenberg 簇几何的开放性问题，但也引发了关于这些事实的表示理论原因的进一步问题。
英文摘要
We study the type A nilpotent Hessenberg varieties associated with Hessenberg functions that satisfy $h(i)\le i+1$. We call these the generalized parabolic Peterson varieties. We show that such varieties can be decomposed into the union of specific generalized parabolic Peterson varieties $\operatorname{Pet}_{λ,α}$, such that $λ$ is an integer partition, $α$ is an integer composition, and $α$ is dominated by $λ$. We prove that when $α$ is dominated by $λ$, the cardinality of the maximal dimensional components of $\operatorname{Pet}_{λ,α}$ equals the Kostka number $\mathcal{K}_{λα}$, and its dimension is determined only by $λ$ and the length of $α$. We provide a recursive formula for the Poincaré polynomial of $\operatorname{Pet}_{λ,α}$. These results partially answer open questions about the geometry of Hessenberg varieties but raise further questions about the representation-theoretic reasons for these facts.
