Kazhdan–Lusztig functions and Chow functions of partially ordered setsOberseminar Darstellungstheorie

by Jacob Matherne (NC State University)

Friday Jun 20, 2025, 2:00 PM → 3:00 PM Europe/Berlin

1.008 (Mathezentrum)

Three decades ago, Stanley and Brenti initiated the study of the Kazhdan–Lusztig–Stanley (KLS) functions.  Roughly, they showed how to associate a KLS function to any poset, thus putting on common ground three important examples: the classical Kazhdan–Lusztig polynomials of Coxeter groups, the matroidal Kazhdan–Lusztig polynomials, and the toric g-polynomials of polytopes.  In this talk, we will recall the KLS story with a focus on these three important examples.

We will also develop a theory that parallels the KLS theory by associating a so-called Chow function to any poset.  In the three respective examples above, the Chow function enumerates paths in the Bruhat graph according to a descent-like statistic, is the Hilbert series of the Chow ring of a matroid (hence the name "Chow function"), and is the h-polynomial of the barycentric subdivision of the poset.  We will explore the relation between the KLS and Chow functions, and illustrate how one can build upon intuition from one setting (say, polytopes) and use it in another (say, Coxeter groups).  Based on joint works with Luis Ferroni and Lorenzo Vecchi, as well as with Tom Braden, June Huh, Nicholas Proudfoot, and Botong Wang.