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.
