Lower bounds for Kazhdan-Lusztig polynomials from patterns
By Sara C. Billey, Tom Braden. 2002
63 citations recorded by the source corpus.
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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/890dd7060c21d278c97e10e294c0390138d0f579
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In Discrete Mathematics and Combinatorics
- Alexander Yong
- Alexander Woo
- Frank Sottile
- Shawn Robinson
- Cristian Lenart
- Anders Skovsted Buch
- Hugh Thomas
- Matthew J. Samuel
In Geometry and Topology
- Alexander Yong
- Alexander Woo
- Frank Sottile
- Shawn Robinson
- Cristian Lenart
- Marcelo Aguiar
- Li Guo
- Kurusch Ebrahimi-Fard
In Mathematical Physics
- Alexander Yong
- Frank Sottile
- Shawn Robinson
- Cristian Lenart
- Alexander Woo
- Anders Skovsted Buch
- Hugh Thomas
- Matthew J. Samuel
Citations in the corpus
- Grothendieck polynomials via permutation patterns and chains in the Bruhat order (2004, methodology): “This pattern map was studied both by Bergeron and Sottile [2] and by Billey and Braden [4], but from different viewpoints.”
- GOVERNING SINGULARITIES OF SCHUBERT VARIETIES (2006, background): “On the other hand, Billey and Braden [1] have given a geometric explanation of why ordinary pattern avoidance characterizes smoothness.”
- When is a Schubert variety Gorenstein (2004, background): “A geometric explanation was recently given in [3] for the appearance of pattern avoidance in characterizations of smooth Schubert varieties.”