Oriented Temperley--Lieb algebras and combinatorial Kazhdan--Lusztig theory

TL;DR

This paper introduces oriented Temperley--Lieb algebras tailored for classical Hermitian symmetric spaces, providing a combinatorial framework to compute Kazhdan--Lusztig polynomials explicitly.

Contribution

It defines new algebraic structures that facilitate combinatorial formulas for Kazhdan--Lusztig polynomials in specific symmetric spaces.

Findings

Existence of closed combinatorial formulas for Kazhdan--Lusztig polynomials

Introduction of oriented Temperley--Lieb algebras for Hermitian symmetric spaces

Enhanced understanding of algebraic structures underlying these polynomials

Abstract

We define oriented Temperley--Lieb algebras for classical Hermitian symmetric spaces. This allows us to explain the existence of closed combinatorial formulae for the Kazhdan--Lusztig polynomials for these spaces.