42 research outputs found
On inversion sets and the weak order in Coxeter groups
In this article, we investigate the existence of joins in the weak order of
an infinite Coxeter group W. We give a geometric characterization of the
existence of a join for a subset X in W in terms of the inversion sets of its
elements and their position relative to the imaginary cone. Finally, we discuss
inversion sets of infinite reduced words and the notions of biconvex and
biclosed sets of positive roots.Comment: 22 pages; 10 figures; v2 some references were added; v2: final
version, to appear in European Journal of Combinatoric
Bruhat order, smooth Schubert varieties, and hyperplane arrangements
The aim of this article is to link Schubert varieties in the flag manifold
with hyperplane arrangements. For a permutation, we construct a certain
graphical hyperplane arrangement. We show that the generating function for
regions of this arrangement coincides with the Poincare polynomial of the
corresponding Schubert variety if and only if the Schubert variety is smooth.
We give an explicit combinatorial formula for the Poincare polynomial. Our main
technical tools are chordal graphs and perfect elimination orderings.Comment: 14 pages, 2 figure
The Order Dimension of the Poset of Regions in a Hyperplane Arrangement
We show that the order dimension of the weak order on a Coxeter group of type
A, B or D is equal to the rank of the Coxeter group, and give bounds on the
order dimensions for the other finite types. This result arises from a unified
approach which, in particular, leads to a simpler treatment of the previously
known cases, types A and B. The result for weak orders follows from an upper
bound on the dimension of the poset of regions of an arbitrary hyperplane
arrangement. In some cases, including the weak orders, the upper bound is the
chromatic number of a certain graph. For the weak orders, this graph has the
positive roots as its vertex set, and the edges are related to the pairwise
inner products of the roots.Comment: Minor changes, including a correction and an added figure in the
proof of Proposition 2.2. 19 pages, 6 figure
Sortable Elements for Quivers with Cycles
Each Coxeter element c of a Coxeter group W defines a subset of W called the
c-sortable elements. The choice of a Coxeter element of W is equivalent to the
choice of an acyclic orientation of the Coxeter diagram of W. In this paper, we
define a more general notion of Omega-sortable elements, where Omega is an
arbitrary orientation of the diagram, and show that the key properties of
c-sortable elements carry over to the Omega-sortable elements. The proofs of
these properties rely on reduction to the acyclic case, but the reductions are
nontrivial; in particular, the proofs rely on a subtle combinatorial property
of the weak order, as it relates to orientations of the Coxeter diagram. The
c-sortable elements are closely tied to the combinatorics of cluster algebras
with an acyclic seed; the ultimate motivation behind this paper is to extend
this connection beyond the acyclic case.Comment: Final version as published. An error corrected in the previous
counterexample, other minor improvement
The freeness of Ish arrangements
International audienceThe Ish arrangement was introduced by Armstrong to give a new interpretation of the -Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be freeL’arrangement Ish a été introduit par Armstrong pour donner une nouvelle interprétation des nombres -Catalan de Garsia et Haiman. Armstrong et Rhoades ont montré qu’il y avait des ressemblances frappantes entre l’arrangement Shi et l’arrangement Ish et ont posé des conjectures. L’une d’elles est de savoir si l’arrangement Ish est un arrangement libre ou pas. Dans cet article, nous vérifions que l’arrangement Ish est supersoluble et donc libre. De plus, on donne une condition nécessaire et suffisante pour que l’arrangement Ish réduit soit libre