Diagrammatics for Kazhdan-Lusztig R-polynomials

TL;DR

This paper introduces a diagrammatic method to define and compute a family of polynomials related to Kazhdan-Lusztig R-polynomials for Coxeter systems, providing new formulas and insights into their structure.

Contribution

The paper develops a diagrammatic framework for Kazhdan-Lusztig R-polynomials, linking them to light leaves and deriving explicit formulas for these polynomials.

Findings

The polynomials $ ilde{ }$ coincide with Kazhdan-Lusztig $ ilde{R}$-polynomials for reduced expressions.

Diagrammatic approach yields closed-form formulas for $ ilde{ }$-polynomials.

New connections between light leaves and R-polynomials established.

Abstract

Let $(W,S)$ be an arbitrary Coxeter system. We introduce a family of polynomials, $\{ \tilde{\mathcal{R}}_{u,\underline{v}}(t)\}$, indexed by pairs $(u,\underline{v})$ formed by an element $u\in W$ and a (non-necessarily reduced) word $\underline{v}$ in the alphabet $S$. The polynomial $\tilde{\mathcal{R}}_{u,\underline{v}}(t)$ is obtained by considering a certain subset of Libedinsky's light leaves associated to the pair $(u,\underline{v})$. Given a reduced expression $\underline{v}$ of an element $v\in W$; we show that $\tilde{\mathcal{R}}_{u,\underline{v}}(t)$ coincides with the Kazhdan-- Lusztig $\tilde{R}$-polynomial $\tilde{R}_{u,v}(t)$. Using the diagrammatic approach, we obtain some closed formulas for $\tilde{R}$- polynomials.