Kazhdan-Lusztig Polynomials for 321-Hexagon-Avoiding Permutations

By Sara C. Billey, Gregory S. Warrington. 2000

99 citations recorded by the source corpus.

AbstractIn (Deodhar, Geom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the Kazhdan-Lusztig polynomials Px,w in the case where W is any Coxeter group. We explicitly describe the combinatorics in the case where $$W = \mathfrak{S}_n $$ (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 Xw to have a small resolution. We conclude with a simple method for completely determining the singular locus of Xw when w is 321-hexagon-avoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points (BCn, F4, G2).

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2a55f77a051e36fb817bcec9533ea7456671d81b

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Citations in the corpus

Kazhdan–Lusztig immanants and products of matrix minors (2006, background): “The Temperley-Lieb algebra may be realized as a quotient of the Hecke algebra by Hn(q)/(z[1,3]) ∼= TLn(q + q−1/2), where the element z[1,3] of Hn(q) is defined as before Proposition 2.”
Maximal singular loci of Schubert varieties in $SL(n)/B$ (2001, background): “Lascoux and Schützenberger [24], Zelevinskĭı [31], Lascoux [23], Brenti [6, 7, 8] and others [5, 9, 28] all calculate explicit formulas for these polynomials in specific cases.”
On the dual canonical and Kazhdan-Lusztig bases and 3412-, 4231-avoiding permutations (2008, background): “The following result of Billey and Warrington [2] uses pattern avoidance to characterize basis elements which factor as products of these.”
Lower bounds for Kazhdan-Lusztig polynomials from patterns (2002, background): “expression w = sa1sa2 ··· sap if and only if w is 321-hexagon-avoiding, [ BiW1 ].”
Parabolic category O, perverse sheaves on Grassmannians, Springer fibres and Khovanov homology (2006, unknown): “by [ 7, Theorem 1 ], the self-adjointness of θs and [40, Section 3]. On the other hand, if ws ∈ W p, ws > w then obviously G W(˜ av)˜ aw � = G W(˜ avs)˜ aws � which is zero if ws / ∈ W p. If ws ∈ W p and ws < w then G W(˜ av)˜ aws � differs from G W(˜ av)˜ aw � by a black circle.”
A New Class of Wilf-Equivalent Permutations (2001, background): “And Billey and Warrington [ 3 ] have very recently defined a class of permutations under 5 restrictions which are related to the Kazhdan-Lusztig polynomials.”