A Class of Kazhdan-Lusztig R-Polynomials and q-Fibonacci Numbers
William Y.C. Chen, Neil J.Y. Fan, Peter L. Guo, and Michael X.X. Zhong
Abstract: Let S_{n} denote the symmetric group on {1, 2,..., n}. For two permutations u, v ∈ S_{n} such that u ≤ v in the Bruhat order, let R_{u,v}(q) and denote the Kazhdan-Lusztig R-polynomial and polynomial, respectively. Let v_{n} = 34···n12, and let σ be a permutation such that σ ≤ v_{n}. We obtain a formula for the -polynomials _{σ,vn}(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 polynomial _{e, vn,i}(q), where v_{n,i} = 34···in(i + 1)···(n - 1)12. In a more general context, we conjecture that for any two permutations σ, τ ∈ S_{n} such that σ ≤ τ ≤ v_{n}, the -polynomial _{σ,τ}(q) can be expressed as a product of q-Fibonacci numbers multiplied by a power of q. AMS Classification: 05E15, 20F55 Keywords: Kazhdan-Lusztig R-polynomial, q-Fibonacci number, symmetric group Download: pdf |