Combinatorial Proof of the Inversion Formula on the Kazhdan-Lusztig R-Polynomials

William Y. C. Chen, Neil J.Y. Fan, Alan J.X. Guo, Peter L. Guo, Harry H.Y. Huang and Michael X.X. Zhong

  Abstract:   In this paper, we present a combinatorial proof of the inversion formula on the Kazhdan-Lusztig R-polynomials. This problem was raised by Brenti. As a consequence, we obtain a combinatorial interpretation of the equi-distribution property due to Verma stating that any nontrivial interval of a Coxeter group in the Bruhat order has as many elements of even length as elements of odd length. The same argument leads to a combinatorial proof of an extension of Verma's equi-distribution to the parabolic quotients of a Coxeter group obtained by Deodhar. As another application, we derive a refinement of the inversion formula for the symmetric group by restricting the summation to permutations ending with a given element.

  AMS Classification:  05A19, 05E15, 20F55

  Keywords:   Kazhdan-Lusztig R-polynomial, inversion formula, Bruhat order

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