Geodesic
Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations
Journal of Algebraic Combinatorics, Tome 13 (2001) no. 2, pp. 111-136.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: In (Deodhar, Geom. Dedicata, $36(1) (1990)$, 95-119), Deodhar proposes a combinatorial framework for determining the Kazhdan-Lusztig polynomials $P _{ x }, _{ w }$ in the case where $W$ is any Coxeter group. We explicitly describe the combinatorics in the case where $W = \mathfrak S _{ n}$ W = mathfrakS_n (the symmetric group on $n$ letters) and the permutation $w$ is 321-hexagon-avoiding. Our formula can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for $w$. As a consequence of our results on Kazhdan-Lusztig polynomials, we show that the Poincaré polynomial of the intersection cohomology of the Schubert variety corresponding to $w$ is $(1+ q) ^{ l(w) }$ if and only if $w$ is 321-hexagon-avoiding. We also give a sufficient condition for the Schubert variety $X _{ w }$ to have a small resolution. We conclude with a simple method for completely determining the singular locus of $X _{ w }$ when $w$ is 321-hexagon-avoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points $( B C _{ n }, F _{4}, G _{2})$.
Keywords: 321-hexagon-avoiding, Kazhdan-Lusztig polynomials, Schubert varieties, singular locus, defect graph 
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     author = {Billey, Sara C. and Warrington, Gregory S.},
     title = {Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations},
     journal = {Journal of Algebraic Combinatorics},
     pages = {111--136},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2001__13_2_a6/}
}
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Billey, Sara C.; Warrington, Gregory S. Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations. Journal of Algebraic Combinatorics, Tome 13 (2001) no. 2, pp. 111-136. http://geodesic.mathdoc.fr/item/JAC_2001__13_2_a6/