Geodesic
Kazhdan-Lusztig polynomials of Boolean elements
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 2, pp. 497-525.

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We give closed combinatorial product formulas for Kazhdan-Lusztig polynomials and their parabolic analogue of type $q$ in the case of Boolean elements, introduced in [M. Marietti, J. Algebra 295, No. 1, 1--26 (2006; Zbl 1097.20035)], 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.
Classification : 05E15, 20F55, 14N15, 20C08
Keywords: Coxeter groups, Kazhdan-Lusztig polynomials, Boolean elements, Poincaré polynomials 
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     author = {Mongelli, Pietro},
     title = {Kazhdan-Lusztig polynomials of {Boolean} elements},
     journal = {Journal of Algebraic Combinatorics},
     pages = {497--525},
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     number = {2},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2014__39_2_a0/}
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Mongelli, Pietro. Kazhdan-Lusztig polynomials of Boolean elements. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 2, pp. 497-525. http://geodesic.mathdoc.fr/item/JAC_2014__39_2_a0/