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.
Billera, L., Brenti, F. (2011). Quasisymmetric functions and Kazhdan-Lusztig polynomials. ISRAEL JOURNAL OF MATHEMATICS, 184(1), 317-348 [10.1007/s11856-011-0070-0].
Quasisymmetric functions and Kazhdan-Lusztig polynomials
BRENTI, FRANCESCO
2011-01-01
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.File | Dimensione | Formato | |
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