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