35 research outputs found
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
Root polytopes and Borel subalgebras
Let be a finite crystallographic irreducible root system and be the convex hull of the roots in . We give a uniform explicit
description of the polytope , analyze the
algebraic-combinatorial structure of its faces, and provide connections with
the Borel subalgebra of the associated Lie algebra. We also give several
enumerative results.Comment: revised version, accepted for publication in IMR
Polar Root Polytopes that are Zonotopes
Let be the root polytope of a finite irreducible
crystallographic root system , i.e., the convex hull of all roots in
. The polar of , denoted ,
coincides with the union of the orbit of the fundamental alcove under the
action of the Weyl group. In this paper, we establishes which polytopes
are zonotopes and which are not. The proof is
constructive.Comment: 12 page
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
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
Pircon kernels and up-down symmetry
We show that a symmetry property that we call the up-down symmetry implies
that the Kazhdan--Lusztig -polynomials of a pircon are a -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
IS THERE A LINK BETWEEN PERIAMPULLARY DIVERTICULA AND BILIOPANCREATIC DISEASES? AN EUS APPROACH TO ANSWER THE QUESTION
Root polytopes and abelian ideals
We study the root polytope of a finite irreducible
crystallographic root system using its relation with the abelian ideals
of a Borel subalgebra of a simple Lie algebra with root system . We
determine the hyperplane arrangement corresponding to the faces of codimension
2 of and analyze its relation with the facets of . For of type or , we show that the orbits of some
special subsets of abelian ideals under the action of the Weyl group
parametrize a triangulation of . We show that this
triangulation restricts to a triangulation of the positive root polytope
.Comment: 41 pages, revised version, accepted for publication in Journal of
Algebraic Combinatoric