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 equidistribution 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 equidistribution 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.
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Acknowledgments
We wish to thank the referee for valuable suggestions. This work was supported by the 973 Project and the National Science Foundation of China.
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Chen, W.Y.C., Fan, N.J.Y., Guo, A.J.X. et al. Combinatorial proof of the inversion formula on the Kazhdan–Lusztig \(R\)-polynomials. Math. Z. 277, 1017–1025 (2014). https://doi.org/10.1007/s00209-014-1291-9
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DOI: https://doi.org/10.1007/s00209-014-1291-9