Peak algebras, paths in the Bruhat graph and Kazhdan-Lusztig polynomials

TL;DR

This paper introduces a new basis for the peak subalgebra of quasisymmetric functions and provides a simplified, general combinatorial formula for Kazhdan-Lusztig polynomials, demonstrating its optimal complexity.

Contribution

It offers a novel characterization and basis for the peak subalgebra, along with a simplified, universal combinatorial formula for Kazhdan-Lusztig polynomials.

Findings

New basis for the peak subalgebra of quasisymmetric functions

A simplified, explicit combinatorial formula for Kazhdan-Lusztig polynomials

Proof that the formula cannot be further simplified in a certain sense

Abstract

We give a new characterization of the peak subalgebra of the algebra of quasisymmetric functions and use this to construct a new basis for this subalgebra. As an application of these results we obtain a combinatorial formula for the Kazhdan-Lusztig polynomials which holds in complete generality and is simpler and more explicit than any existing one. We then show that, in a certain sense, this formula cannot be simplified.