AI 中文总结
研究如何确定旗簇中黑森伯格簇的闭子簇结构，通过对黑森伯格函数施加偏序并分析相关簇关系，给出在舒伯特胞腔中确定切出给定黑森伯格簇方程的递归公式，还为泰莫茨科的相关结果提供几何证明。
AI 中文摘要
黑森伯格簇是旗簇的子簇，由关于线性算子的旗的包含条件定义。对这些簇的研究涉及代数几何、组合学和表示理论的交叉。本文通过对黑森伯格函数施加偏序并分析相应黑森伯格簇的关系，开发了一种代数几何程序来确定对于任何线性算子\(X\)和黑森伯格函数\(h\)的旗簇中黑森伯格簇\(\mathcal{H}(X,h)\)的闭子簇结构。特别地，给出了在每个舒伯特胞腔中确定切出给定黑森伯格簇的所有方程的具体递归公式。作为应用，为泰莫茨科关于给定黑森伯格簇仿射铺砌的存在性及其胞腔维数计数的结果提供了另一种几何证明。
英文摘要
Hessenberg varieties are subvarieties of the flag variety, defined by containment conditions on flags with respect to a linear operator. The study of these varieties lies in the intersection of algebraic geometry, combinatorics, and representation theory. In this paper, we develop an algebro-geometric procedure for determining the closed subvariety structure of a Hessenberg variety $\mathcal{H}(X,h)$ in the flag variety for any linear operator $X$ and Hessenberg function $h$, by imposing a partial order on the Hessenberg functions and analyzing the relation of the corresponding Hessenberg varieties. In particular, we give a concrete recursive formula for determining all equations cutting out a given Hessenberg variety in each Schubert cell. As an application, we provide an alternative geometric proof of Tymoczko's results on the existence of affine pavings of a given Hessenberg variety and on the dimension count of its cells.
Comments24 pages, 8 figures, 1 table
