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 Sn denote the symmetric group on {1, 2,..., n}. For two permutations u, v ∈ Sn such that u ≤ v in the Bruhat order, let Ru,v(q) and denote the Kazhdan-Lusztig R-polynomial and polynomial, respectively. Let vn = 34···n12, and let σ be a permutation such that σ ≤ vn. 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 vn,i = 34···in(i + 1)···(n - 1)12. In a more general context, we conjecture that for any two permutations σ, τ ∈ Sn such that σ ≤ τ ≤ vn, 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

  Kazhdan-Lusztig R-polynomial, q-Fibonacci number, symmetric group

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