2 research outputs found

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

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    Let SnS_n denote the symmetric group on {1,2,…,n}\{1,2,\ldots,n\}. For two permutations u,v∈Snu, v\in S_n such that u≀vu\leq v in the Bruhat order, let Ru,v(q)R_{u,v}(q) and Ru,v(q)\R_{u,v}(q) denote the Kazhdan-Lusztig RR-polynomial and R\R-polynomial, respectively. Let vn=34β‹―n 12v_n=34\cdots n\, 12, and let Οƒ\sigma be a permutation such that σ≀vn\sigma\leq v_n. We obtain a formula for the R\R-polynomials RΟƒ,vn(q)\R_{\sigma,v_n}(q) in terms of the qq-Fibonacci numbers depending on a parameter determined by the reduced expression of Οƒ\sigma. When Οƒ\sigma is the identity ee, this reduces to a formula obtained by Pagliacci. In another direction, we obtain a formula for the R\R-polynomial Re, vn,i(q)\R_{e,\,v_{n,i}}(q), where vn,i=34β‹―i n (i+1)β‹―(nβˆ’1) 12v_{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 Οƒ,Ο„βˆˆSn\sigma,\tau\in S_n such that σ≀τ≀vn\sigma\leq \tau\leq v_n, the R\R-polynomial RΟƒ,Ο„(q)\R_{\sigma,\tau}(q) can be expressed as a product of qq-Fibonacci numbers multiplied by a power of qq.Comment: 11 page

    Diagrammatics for Kazhdan-Lusztig R-polynomials

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    Let (W,S)(W,S) be an arbitrary Coxeter system. We introduce a family of polynomials, {R~u,vβ€Ύ(t)}\{ \tilde{\mathcal{R}}_{u,\underline{v}}(t)\}, indexed by pairs (u,vβ€Ύ)(u,\underline{v}) formed by an element u∈Wu\in W and a (non-necessarily reduced) word vβ€Ύ\underline{v} in the alphabet SS. The polynomial R~u,vβ€Ύ(t)\tilde{\mathcal{R}}_{u,\underline{v}}(t) is obtained by considering a certain subset of Libedinsky's light leaves associated to the pair (u,vβ€Ύ)(u,\underline{v}). Given a reduced expression vβ€Ύ\underline{v} of an element v∈Wv\in W; we show that R~u,vβ€Ύ(t)\tilde{\mathcal{R}}_{u,\underline{v}}(t) coincides with the Kazhdan-- Lusztig R~\tilde{R}-polynomial R~u,v(t)\tilde{R}_{u,v}(t). Using the diagrammatic approach, we obtain some closed formulas for R~\tilde{R}- polynomials.Comment: 20 pages, best viewed in colo
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