Abstract
Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations $w$ with Kazhdan-Lusztig polynomial $P_{id,w}(q)=1+q^h$ for some $h$. (Appendix by Sara Billey and Jonathan Weed)
Highlights
Kazhdan-Lusztig polynomials are polynomials Pu,w(q) in one variable associated to each pair of elements u and w in the symmetric group Sn
They have an elementary definition in terms of the Hecke algebra [24, 21, 9] and numerous applications in representation theory, most notably in [24, 1, 13], and the geometry of homogeneous spaces [25, 17]
For Sn, positive formulas are known only for 3412 avoiding permutations [27, 28], 321-hexagon avoiding permutations [7], and some isolated cases related to the generic singularities of Schubert varieties [8, 31, 16, 34]
Summary
Kazhdan-Lusztig polynomials are polynomials Pu,w(q) in one variable associated to each pair of elements u and w in the symmetric group Sn (or more generally in any Coxeter group). That Pid,w(1) ≤ 2 implies the pattern avoidance conditions follows from [6, Thm. 1] and the computation of Kazhdan-Lusztig polynomials for the six pattern permutations. While this paper was being written, Billey and Weed found an alternative formulation of Theorem 1.1 purely in terms of pattern avoidance, replacing the condition that the singular locus of Xw have only one component with sixty patterns. To prove Theorem 1.1, we study resolutions of singularities for Schubert varieties that were introduced by Cortez [15, 16] and use an interpretation of the Decomposition Theorem [2] given by Polo [32] which allows computation of Kazhdan-Lusztig polynomials Pv,w (and more generally local intersection homology Poincare polynomials for appropriate varieties) from information about the fibers of a resolution of singularities. I used Greg Warrington’s software [33] for computing Kazhdan-Lusztig polynomials in explorations leading to this work
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