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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
Article

Abstract

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.

Keywords

Representation Theory Topological Group Symmetric Group Weyl Group Decomposition Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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