Special matchings and parabolic Kazhdan-Lusztig polynomials

TL;DR

This paper introduces a combinatorial approach using special matchings to compute parabolic Kazhdan-Lusztig polynomials for certain Coxeter groups, extending previous results and highlighting the importance of Bruhat interval structures.

Contribution

It demonstrates that special matchings can be used to compute parabolic Kazhdan-Lusztig polynomials in doubly laced and dihedral Coxeter groups, generalizing prior work.

Findings

Parabolic Kazhdan-Lusztig polynomials depend only on Bruhat interval poset structure.

Results apply to all Weyl groups and similar Coxeter groups.

The approach simplifies computation of these polynomials.

Abstract

We prove that the combinatorial concept of a special matching can be used to compute the parabolic Kazhdan-Lusztig polynomials of doubly laced Coxeter groups and of dihedral Coxeter groups. In particular, for this class of groups which includes all Weyl groups, our results generalize to the parabolic setting the main results in [Advances in Math. {202} (2006), 555-601]. As a consequence, the parabolic Kazhdan-Lusztig polynomial indexed by $u$ and $v$ depends only on the poset structure of the Bruhat interval from the identity element to $v$ and on which elements of that interval are minimal coset representatives.