A Generalization of Deodhar's Framework for Questions in Kazhdan-Lusztig Theory

Abstract: We make progress on a question of Skandera by showing that a product of Kazhdan-Lusztig basis elements indexed by maximal elements of parabolic subgroups admits a Kazhdan-Lusztig basis element as a quotient arising from operations in the Schur algebroid if and only if the sequence of parabolic subgroups satisfy both a rigidity condition and a combinatorial criterion. For Weyl groups, the rigidity condition specializes to a necessary condition for smallness of Gelfand-MacPherson resolutions. For Schubert varieties indexed by 4231-avoiding permutations, we derive a stronger necessary condition that, up to an appropriate equivalence, is satisfied by at most one Gelfand-MacPherson resolution, and exactly one if and only if 45312 is also avoided. Moreover, we apply the combinatorial criterion to prove the (essentially unique) resolution is small when 34512 and 45123 are likewise avoided. We develop the combinatorial criterion as part of a generalization of Deodhar's combinatorial setting for questions in Kazhdan-Lusztig theory, which in the case of Weyl groups we show captures the Bya{\l}ynicki-Birula decompositions of Gelfand-MacPherson resolutions. In particular, we obtain new combinatorial interpretations of Kazhdan-Lusztig polynomials, new algorithms for computing them, and in the case of Weyl groups, an equivalence of the existence of small Gelfand-MacPherson resolutions with that of certain factorizations and generating-function interpretations of the Poincar\'e polynomial of intersection cohomology of the Schubert variety.