A Class of Kazhdan-Lusztig R-Polynomials and q-Fibonacci Numbers

Abstract

Let $S_n$ denote the symmetric group on $\{1,2,\dots ,n\}$. For two permutations $u,v\in S_n$ such that $u\le v$ in the Bruhat order, let $R_{u,v}(q)$ and ${\mathbb{R}}_{u,v}(q)$ denote the Kazhdan-Lusztig $R$-polynomial and $\mathbb{R}$-polynomial, respectively. Let $v_n=34\cdots n12$, and let $\sigma$ be a permutation such that $\sigma \le v_n$. We obtain a formula for the $\mathbb{R}$-polynomials ${\mathbb{R}}_{\sigma ,v_n}(q)$ in terms of the $q$-Fibonacci numbers depending on a parameter determined by the reduced expression of $\sigma$. When $\sigma$ is the identity $e$, this reduces to a formula obtained by Pagliacci. In another direction, we obtain a formula for the $\mathbb{R}$-polynomial ${\mathbb{R}}_{e,v_{n,i}}(q)$, where $v_{n,i}=34\cdots in(i+1)\cdots (n-1)12$. In a more general context, we conjecture that for any two permutations $\sigma ,\tau \in S_n$ such that $\sigma \le \tau \le v_n$, the $\mathbb{R}$-polynomial ${\mathbb{R}}_{\sigma ,\tau }(q)$ can be expressed as a product of $q$-Fibonacci numbers multiplied by a power of $q$.

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