A simple characterization of special matchings in lower Bruhat intervals

We give a simple characterization of special matchings in lower Bruhat intervals (that is, intervals starting from the identity element) of a Coxeter group. As a byproduct, we obtain some results on the action of special matchings.Comment: accepted for publication on Discrete Mathematic

A simple characterization of special matchings in lower Bruhat intervals

We give a simple characterization of special matchings in lower Bruhat intervals (that is, intervals starting from the identity element) of a Coxeter group. As a byproduct, we obtain some results on the action of special matchings.Comment: accepted for publication on Discrete Mathematic

Special matchings in Coxeter groups

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.Comment: 19 page

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

The aim of this work is to prove a conjecture related to the Combinatorial Invariance Conjecture of Kazhdan-Lusztig polynomials, in the parabolic setting, for lower intervals in every arbitrary Coxeter group. This result improves and generalizes, among other results, the main results of [Advances in Math. {202} (2006), 555-601], [Trans. Amer. Math. Soc. {368} (2016), no. 7, 5247--5269].Comment: to appear in Advances in Mathematic

Pircon kernels and up-down symmetry

We show that a symmetry property that we call the up-down symmetry implies that the Kazhdan--Lusztig $R^x$-polynomials of a pircon $P$ are a $P$-kernel, and we show that this property holds in the classical cases. Then, we enhance and extend to this context a duality of Deodhar in parabolic Kazhdan--Lusztig theory.Comment: to appear in Journal of Algebra. arXiv admin note: substantial text overlap with arXiv:1907.0085

Special idempotents and projections

We define, for any special matching of a finite graded poset, an idempotent, regressive and order preserving function. We consider the monoid generated by such functions. The idempotents of this monoid are called special idempotents. They are interval retracts. Some of them realize a kind of parabolic map and are called special projections. We prove that, in Eulerian posets, the image of a special projection, and its complement, are graded induced subposets. In a finite Coxeter group, all projections on right and left parabolic quotients are special projections, and some projections on double quotients too. We extend our results to special partial matchings

Combinatorial invariance for elementary intervals

We adapt the hypercube decompositions introduced by Blundell-Buesing-Davies-Veli\v{c}kovi\'{c}-Williamson to prove the Combinatorial Invariance Conjecture for Kazhdan-Lusztig polynomials in the case of elementary intervals in $S_n$. This significantly generalizes the main previously-known case of the conjecture, that of lower intervals.Comment: 15 pages, comments welcom