Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Diagrammatics for Kazhdan-Lusztig R-polynomials   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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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: patt-sol
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: funct-an
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The categories of this paper are: acc-phys
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The categories of this paper are: q-bio
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The categories of this paper are: stat
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The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas