长程差商、簇与Graham正性
Long range divided differences, clusters, and Graham-positivity
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中文总结 AI 辅助
本文通过长程差商运算和组合算法，将环面轨道闭包的等变同调类展开为Graham正Schubert循环组合，推广了AJS-Billey公式并应用于曲线、轨道闭包及Richardson簇。
中文摘要 AI 辅助
我们研究了正簇扇中与锥自然关联的$A$型完全旗簇中的环面轨道闭包及其左$S_n$-平移。环面等变度映射可通过由非交叉交错森林编码的长程差商运算的复合来计算，我们给出了组合算法，将环面等变同调类展开为Schubert循环的Graham正组合。作为应用，我们获得了所有环面不变曲线（推广了环面不动点的AJS-Billey公式）、一般环面轨道闭包以及Bruhat区间$[w,wc']$（其中$c'\le s_{n-1}s_{n-2}\cdots s_1$）的Richardson簇的组合Graham正Schubert循环展开。投影到Grassmannian，我们还获得了与置换基集上的格路拟阵相关联的环面轨道闭包的Graham正Grassmannian Schubert循环分解。
英文摘要
We study torus-orbit closures in the type $A$ complete flag variety naturally associated to cones in the positive cluster fan, together with their left $S_n$-translates. The torus-equivariant degree maps can be computed via composites of long-range divided difference operations encoded by noncrossing alternating forests, and we give combinatorial algorithms to expand the torus-equivariant homology classes into Graham-positive combinations of Schubert cycles. As applications we obtain combinatorial Graham-positive Schubert cycle expansions for all torus-invariant curves (generalizing the AJS-Billey formula for torus-fixed points), generic torus-orbit closures, and Richardson varieties for Bruhat intervals $[w,wc']$ where $c'\le s_{n-1}s_{n-2}\cdots s_1$. Projecting to Grassmannians we also obtain Graham-positive Grassmannian Schubert cycle decompositions of torus-orbit closures associated to lattice path matroids on permuted ground sets.
发表机构
- University of Toronto（多伦多大学）
- University of Toronto Mississauga（多伦多大学密西沙加分校）
机构由 AI 辅助整理，请以论文原文为准。
