Lower Bounds for Kazhdan-lusztig Polynomials from Patterns
نویسندگان
چکیده
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. In particular, the value Px,w(1) is the dimension of the intersection cohomology sheaf of the Schubert variety Xw at the T -fixed point indexed by x [KL2]. It is also the multiplicity of a certain irreducible module in a Verma module [BeBe][BryK]. In this paper, we give a lower bound for the values Px,w(1) in terms of “patterns”. These patterns correspond with special subgroups of the Weyl group generated by reflections. This notion generalizes the concept of patterns and pattern avoidance for permutations to all Weyl groups. The main tool of the proof is Braden’s ”hyperbolic localization” on intersection cohomology. This construction gives some insight into the geometry behind pattern avoidance.
منابع مشابه
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