Quasisymmetric functions and Kazhdan-Lusztig polynomials

Details Quasisymmetric functions and Kazhdan-Lusztig polynomials Quasisymmetric functions and Kazhdan-Lusztig polynomials

2007-10-22

arxiv.org/abs/0710.3965v2

Mathematics

Combinatorics

27 pages. Final version: definitions reorganized for clarity, added Example 4.6 and two citations. To appear in Israel Journal

Scientific paper

We associate a quasisymmetric function to any Bruhat interval in a general Coxeter group. This association can be seen to be a morphism of Hopf algebras to the subalgebra of all peak functions, leading to an extension of the cd-index of convex polytopes. We show how the Kazhdan-Lusztig polynomial of the Bruhat interval can be expressed in terms of this complete cd-index and otherwise explicit combinatorially defined polynomials. In particular, we obtain the simplest closed formula for the Kazhdan-Lusztig polynomials that holds in complete generality.

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