Special matchings in Coxeter groups

TL;DR

This paper provides a complete classification of special matchings in Coxeter groups, offering new combinatorial tools for understanding Bruhat intervals and their applications in Kazhdan--Lusztig polynomials.

Contribution

It offers an explicit characterization and classification of all special matchings in any lower Bruhat interval within Coxeter groups.

Findings

Complete classification of special matchings in Coxeter groups

Applicable to all lower Bruhat intervals

Implications for parabolic Kazhdan--Lusztig polynomials

Abstract

Special matchings are purely combinatorial objects associated with a partially ordered set, which have applications in Coxeter group theory. We provide an explicit characterization and a complete classification of all special matchings of any lower Bruhat interval. The results hold in any arbitrary Coxeter group and have also applications in the study of the corresponding parabolic Kazhdan--Lusztig polynomials.