Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Generalization of Deodhar's Framework for Questions in Kazhdan-Lusztig
  Theory   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.

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The categories of this paper are: hep-ex
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: eess
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: cs
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The categories of this paper are: stat
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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph