Combinatorial Representation Theory: Webs and Kazhdan--Lusztig Polynomials

This project focuses on two big open problems in algebraic combinatorics that lie at the intersection of representation theory and algebraic geometry. Both problems have connections to multiple fields of mathematics, and in both cases the PI and his collaborators aim to use combinatorics to understand objects which are currently understood only abstractly in a more concrete way. It is hoped that such understanding will have applications back to the other areas of mathematics, but also to physics and to human-machine collaboration in mathematics. The project will also help develop the STEM workforce by providing research and conference travel support for graduate students. More specifically, the project aims to (1) improve upon recent progress the PI has made on the Combinatorial Invariance Conjecture for Kazhdan-Lusztig polynomials, which asserts that these polynomials of foundational importance in geometric representation theory depend only on the structure of a certain graph, and (2) extend the PI’s recent construction of an SL(4) web basis to produce analogous bases for spaces of invariants of higher rank Lie (or quantum) groups. For (1), the proposed approach takes advantage of the combinatorics of structures called hypercube decompositions of Bruhat intervals, which were discovered with the aid of machine learning. For (2), the PI hopes to deepen the recently discovered connections between webs and the combinatorics of plabic graphs. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.