Jeremy Meza: Hecke algebras and Kazhdan-Lusztig polynomials

Jeremy Meza: Hecke algebras and Kazhdan-Lusztig polynomials.

Abstract

In their seminal 1979 paper, Kazhdan and Lusztig introduced a collection of polynomials for any Coxeter group that have (surprising?) connections to a myriad of topics in algebra, combinatorics, and geometry. We will attempt to survey this program, starting from the Hecke algebra, working our way through to computing Kazhdan-Lusztig polynomials, and ending at the construction of the representations of the associated Coxeter system. We will then specialize to Type A to hopefully see the interactions with already established tableau combinatorics. If time permits, I will mention a modification in Type B, as well as applications of Kazhdan-Lusztig polynomials. No prior knowledge of representation theory is strictly needed.