Transformation Groups

, Volume 8, Issue 4, pp 321–332 | Cite as

Lower bounds for Kazhdan-Lusztig polynomials from patterns

  • Sara C. BilleyEmail author
  • Tom BradenEmail author


Kazhdan-Lusztig polynomials Px,w(q) play an important role in the study of Schubert varieties as well as the representation theory of semisimple Lie algebras. We give a lower bound for the values Px,w(1) in terms of "patterns". A pattern for an element of a Weyl group is its image under a combinatorially defined map to a subgroup generated by reflections. This generalizes the classical definition of patterns in symmetric groups. This map corresponds geometrically to restriction to the fixed point set of an action of a one-dimensional torus on the flag variety of a semisimple group G. Our lower bound comes from applying a decomposition theorem for "hyperbolic localization" [Br] to this torus action. This gives a geometric explanation for the appearance of pattern avoidance in the study of singularities of Schubert varieties.


Representation Theory Topological Group Symmetric Group Weyl Group Decomposition Theorem 
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Copyright information

© Birkhauser Boston 2003

Authors and Affiliations

  1. 1.Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139USA
  2. 2.Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003USA

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