Diagrammatics for Kazhdan-Lusztig R-polynomials

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.

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