443 research outputs found
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
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
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
On the local homology of Artin groups of finite and affine type
We study the local homology of Artin groups using weighted discrete Morse
theory. In all finite and affine cases, we are able to construct Morse
matchings of a special type (we call them "precise matchings"). The existence
of precise matchings implies that the homology has a square-free torsion. This
property was known for Artin groups of finite type, but not in general for
Artin groups of affine type. We also use the constructed matchings to compute
the local homology in all exceptional cases, correcting some results in the
literature
On the top coefficients of Kazhdan-Lusztig polynomials
Kazhdan and Lusztig in 1979 defined, for any Coxeter group W, a family of polynomial that is know as Kazhdan-Lusztig polynomials. These polynomials are indexed by pairs of elements of W and they have fundamental
importance in several areas of mathematics as representation theory, geometry, combinatorics and topology of Schubert varieties. In this paper we want to show the combinatorial connection between special matchings and the top coefficient of Kazhdan-Lusztig polynomial. In particular we study a Conjecture, due to Brenti, in different Coxeter groups and pair of elements
Cluster algebras of type D: pseudotriangulations approach
We present a combinatorial model for cluster algebras of type in terms
of centrally symmetric pseudotriangulations of a regular -gon with a small
disk in the centre. This model provides convenient and uniform interpretations
for clusters, cluster variables and their exchange relations, as well as for
quivers and their mutations. We also present a new combinatorial interpretation
of cluster variables in terms of perfect matchings of a graph after deleting
two of its vertices. This interpretation differs from known interpretations in
the literature. Its main feature, in contrast with other interpretations, is
that for a fixed initial cluster seed, one or two graphs serve for the
computation of all cluster variables. Finally, we discuss applications of our
model to polytopal realizations of type associahedra and connections to
subword complexes and -cluster complexes.Comment: 21 pages, 21 figure
Homotopy Type of the Boolean Complex of a Coxeter System
In any Coxeter group, the set of elements whose principal order ideals are
boolean forms a simplicial poset under the Bruhat order. This simplicial poset
defines a cell complex, called the boolean complex. In this paper it is shown
that, for any Coxeter system of rank n, the boolean complex is homotopy
equivalent to a wedge of (n-1)-dimensional spheres. The number of such spheres
can be computed recursively from the unlabeled Coxeter graph, and defines a new
graph invariant called the boolean number. Specific calculations of the boolean
number are given for all finite and affine irreducible Coxeter systems, as well
as for systems with graphs that are disconnected, complete, or stars. One
implication of these results is that the boolean complex is contractible if and
only if a generator of the Coxeter system is in the center of the group. of
these results is that the boolean complex is contractible if and only if a
generator of the Coxeter system is in the center of the group.Comment: final version, to appear in Advances in Mathematic
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