Abstract
We obtain a nonrecursive combinatorial formula for the Kazhdan-Lusztig polynomials which holds in complete generality and which is simpler and more explicit than any existing one, and which cannot be linearly simplified. Our proof uses a new basis of the peak subalgebra of the algebra of quasisymmetric functions. On montre une formule combinatoire pour les polynômes de Kazhdan-Lusztig qui est valable en toute généralité. Cette formule est plus simple et plus explicite que toutes les autres formules connues; de plus, elle ne peut pas être simplifiée linéairement. La preuve utilise une nouvelle base pour la sous-algèbre des sommets de l’algèbre des fonctions quasi-symmetriques.
Highlights
[19] Kazhdan and Lusztig introduced a family of polynomials, indexed by pairs of elements of a Coxeter group W, that are known as the Kazhdan-Lusztig polynomials of W
These polynomials play a fundamental role in several areas of mathematics, including representation theory, the geometry of Schubert varieties, the theory of Verma modules, Macdonald polynomials, canonical bases, immanant inequalities and the Hodge theory of Soergel bimodules
The purpose of this work is to give a new nonrecursive combinatorial formula for these polynomials which holds in complete generality
Summary
In their seminal paper [19] Kazhdan and Lusztig introduced a family of polynomials, indexed by pairs of elements of a Coxeter group W , that are known as the Kazhdan-Lusztig polynomials of W (see, e.g., [5] or [18]). The purpose of this work is to give a new nonrecursive combinatorial formula for these polynomials which holds in complete generality. This formula expresses the Kazhdan-Lusztig polynomials as a linear combination of polynomials whose coefficients have a simple combinatorial interpretation in terms of a family of lattice paths that we call slaloms. We investigate linear relations arising from the enumeration of Bruhat paths and show, in particular, that the formula that we have obtained cannot be linearly simplified
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