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Kazhdan-Lusztig polynomials of boolean elements

Abstract : We give closed combinatorial product formulas for Kazhdan–Lusztig poynomials and their parabolic analogue of type $q$ in the case of boolean elements, introduced in [M. Marietti, Boolean elements in Kazhdan–Lusztig theory, J. Algebra 295 (2006)], in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincaré polynomials of the Schubert varieties of boolean elements.
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https://hal.inria.fr/hal-01229702
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Pietro Mongelli. Kazhdan-Lusztig polynomials of boolean elements. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.587-598. ⟨hal-01229702⟩

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