Quasisymmetric functions and Kazhdan-Lusztig polynomials
Combinatorics
2009-10-20 v2 Representation Theory
Abstract
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.
Cite
@article{arxiv.0710.3965,
title = {Quasisymmetric functions and Kazhdan-Lusztig polynomials},
author = {Louis J. Billera and Francesco Brenti},
journal= {arXiv preprint arXiv:0710.3965},
year = {2009}
}
Comments
27 pages. Final version: definitions reorganized for clarity, added Example 4.6 and two citations. To appear in Israel Journal of Mathematics