Special matchings calculate the parabolic Kazhdan–Lusztig polynomials of the universal Coxeter groups

Abstract

In this paper we prove that the parabolic Kazhdan-Lusztig polynomials and the parabolic R-polynomials of the universal Coxeter group can be computed in a combinatorial way, by using special matchings.

The combinatorial invariance conjecture for parabolic Kazhdan-Lusztig polynomials of lower intervals

“…Theorems 1.3, 1.4 and 1.5 improve and generalize several results in the literature such as, for instance, the main results of [6], [12], [21], [23].…”

Special matchings in Coxeter groups

“…Since the results in this paper are valid for all Coxeter groups, we believe that they might be useful to extend the results in [11] and [13] also for other classes of Coxeter groups.…”

“…In fact, since the appearance of [4], the authors have been asked many times whether the results in it could be generalized to the parabolic Kazhdan-Lusztig polynomials setting. This problem, still open for a general Coxeter group W , has been recently solved in [11] for the doubly laced Coxeter groups (and, also, in the much easier case of dihedral Coxeter systems) and in [13] for universal Coxeter groups. Both of these papers make use of the classification of special matchings given in the present work.…”

A simple characterization of special matchings in lower Bruhat intervals

“…Over the last few years, special matchings have shown many applications in Coxeter group theory in general (see [11], [12]) and in the computation of Kazhdan-Lusztig polynomials in particular (see [2], [3], [4], [13], [14], [16]).…”