通用舒伯特多项式的簇几何 I：几何基与舒伯特转换
Cluster Geometry of Universal Schubert Polynomials I: Geometric Bases and Schubert Transitions
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中文总结 AI 辅助
该研究将Fulton通用舒伯特多项式作为上幺幂群正则函数，分析其簇结构、几何基及转换关系，证明齐次性条件并展示$S_{10}$中置换对应的三个几何基元素互异。
中文摘要 AI 辅助
我们将Fulton的通用舒伯特多项式$\boldsymbol{\frak S}_w(c)$作为上幺幂群$U_N$上的正则函数进行研究。标准三角簇结构为每个置换关联一个舒伯特$\boldsymbol{\frak g}$-向量，其凸包与$\boldsymbol{\triangle}_1\times\boldsymbol{\triangle}_2\times\boldsymbol{\triangle}_{N-1}$单模等价，其根度纤维为阶梯箭图格拉斯曼层实现的抛物Bruhat区间。由这些向量索引的几何元素（即通用、典范及Mirković--Vilonen元素）构成Fulton标准初等模的整基。我们证明，$\boldsymbol{\frak S}_w(c)$在对角共轭下是齐次的当且仅当它是对应的典范元素，且$\boldsymbol{\frak S}_w(c)$的齐次性蕴含$Z_{\boldsymbol{\frak g}_w}$的$\boldsymbol{\frak \triangle}_Q$-刚性。我们还对几何到舒伯特转换的单位列进行分类，确定PBW到几何及代码到舒伯特转换的支撑分量，并展示了$S_{10}$中一个置换$w$，其对应的三个几何基元素互不相同。
英文摘要
We study Fulton's universal Schubert polynomials $\mathfrak S_w(c)$ as regular functions on the upper unitriangular group $U_N$. The standard triangular cluster structure associates a Schubert $\sf g$-vector to every permutation. Their convex hull is unimodularly equivalent to $Δ_1\times\cdots\timesΔ_{N-1}$, and their root-degree fibers are parabolic Bruhat intervals realized by the strata of staircase quiver Grassmannians. The geometric (i.e., generic, canonical, and Mirković--Vilonen) elements indexed by these vectors form integral bases of Fulton's standard-elementary module. We prove that $\mathfrak S_w(c)$ is homogeneous under diagonal conjugation if and only if it is the corresponding canonical element, and that homogeneity of $\mathfrak S_w(c)$ implies $Λ_Q$-rigidity of $Z_{\sf g_w}$. We also classify simultaneously the unit columns of the geometric-to-Schubert transitions, determine support components of the PBW-to-geometric and code-to-Schubert transitions, and exhibit a permutation $w\in S_{10}$ for which the three geometric basis elements are distinct.
发表机构
- School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
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