arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1312.2170math.CO
收藏

一类Kazhdan-Lusztig R-多项式与q-斐波那契数

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

William Y. C. Chen, Neil J. Y. Fan, Peter L. Guo, Michael X. X. Zhong

备注：
11 pages
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

令$S_n$表示$\{1,2,\ldots,n\}$上的对称群。对于两个排列$u, v\in S_n$使得$u\leq v$为Bruhat阶，令$R_{u,v}(q)$和$\R_{u,v}(q)$分别表示Kazhdan-Lusztig $R$ -多项式和$\R$ -多项式。令$v_n=34\cdots n\, 12$，并令$σ$为置换，使得$σ\leq v_n$。我们得到一个公式的$\R$ -多项式$\R_{σ,v_n}(q)$的$q$ -斐波那契数取决于一个参数所确定的约化表达式的$σ$。当$σ$是恒等式$e$时，这简化为Pagliacci获得的公式。在另一个方向上，我们得到了一个公式的$\R$ -多项式$\R_{e,\,v_{n,i}}(q)$，其中$v_{n,i} = 3 4\cdots i\,n\, (i+1)\cdots (n-1)\, 12$。在更一般的情况下，我们推测，对于任何两个排列$σ,τ\in S_n$，$σ\leq τ\leq v_n$，$\R$ -多项式$\R_{σ,τ}(q)$可以表示为$q$ -斐波那契数乘以$q$的幂的乘积。

英文摘要：

Let $S_n$ denote the symmetric group on $\{1,2,\ldots,n\}$. For two permutations $u, v\in S_n$ such that $u\leq v$ in the Bruhat order, let $R_{u,v}(q)$ and $\R_{u,v}(q)$ denote the Kazhdan-Lusztig $R$-polynomial and $\R$-polynomial, respectively. Let $v_n=34\cdots n\, 12$, and let $σ$ be a permutation such that $σ\leq v_n$. We obtain a formula for the $\R$-polynomials $\R_{σ,v_n}(q)$ in terms of the $q$-Fibonacci numbers depending on a parameter determined by the reduced expression of $σ$. When $σ$ is the identity $e$, this reduces to a formula obtained by Pagliacci. In another direction, we obtain a formula for the $\R$-polynomial $\R_{e,\,v_{n,i}}(q)$, where $v_{n,i} = 3 4\cdots i\,n\, (i+1)\cdots (n-1)\, 12$. In a more general context, we conjecture that for any two permutations $σ,τ\in S_n$ such that $σ\leq τ\leq v_n$, the $\R$-polynomial $\R_{σ,τ}(q)$ can be expressed as a product of $q$-Fibonacci numbers multiplied by a power of $q$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑