Combinatorial Models in Schubert Geometry

PI将研究与Schubert变体和其他重要子变体X相关的重要组合不变量。例子包括著名的Kazhdan-Lusztig多项式和Littlewood-Richardson系数。目标是为这些对象构建组合模型，作为发现共同潜在离散规律的手段。一种分析方法是通过x的“斑块”的几何退化，在平面极限上分解不可约的变种。对分解成分的组合分析揭示了原始变种X.Flag变种及其自然亚变种（如Schubert， Richardson, Peterson，施普林格，k轨道闭包）构成了代数，几何和组合学相互作用的基本基石。本项目旨在扩大和加深我们对这些品种不变量的共同特征的理解。

The PI will investigate important combinatorial invariants related to Schubert varieties and other important subvarieties X of flag manifolds. Examples include the famous Kazhdan-Lusztig polynomials and Littlewood-Richardson coefficients. The goal is to construct combinatorial models for these objects as a means to discover common underlying discrete laws. One technique of analysis is via geometric degeneration of ``patches'' of X. Irreducible varieties are decomposed in the flat limit. Combinatorial analysis of the components of the decomposition informs about the original variety X.Flag varieties and their natural subvarieties (e.g., Schubert, Richardson, Peterson, Springer, K-orbit closures) form a fundamental cornerstone of the interplay between algebra, geometry and combinatorics. This project intends to expand and deepen our understanding of common features of invariants of these varieties.