Grant Barkley
University of Michigan
Title
The combinatorial invariance conjecture for KL polynomials
Abstract
Kazhdan-Lusztig polynomials are a family of polynomials P_uv indexed by two elements u,v in a Coxeter group that encode information about canonical bases of Hecke algebras, singularities of Schubert varieties, and composition factors of Verma modules. Another thing you can associate to a pair u,v is the interval in Bruhat order [u,v]. The combinatorial invariance conjecture of Dyer and Lusztig asserts that P_uv is computable purely from knowledge of [u,v] as an abstract poset. I will talk about this conjecture and recent progress, joint with Christian Gaetz and Thomas Lam. In particular, we have shown that if [u,v] and [u’,v’] are isomorphic as abstract posets, then the coefficients of q in P_uv and P_u’v’ are the same. It turns out proving this depends a lot on understanding the cohomology of Richardson varieties. If time permits, we will also discuss some results in type A based on hypercube decompositions, which are combinatorial structures introduced by Geordie Williamson and Google DeepMind with the help of machine learning.
