20 research outputs found
The absolute order on the hyperoctahedral group
The absolute order on the hyperoctahedral group is investigated. It is
proved that the order ideal of this poset generated by the Coxeter elements is
homotopy Cohen-Macaulay and the M\"obius number of this ideal is computed.
Moreover, it is shown that every closed interval in the absolute order on
is shellable and an example of a non-Cohen-Macaulay interval in the absolute
order on is given. Finally, the closed intervals in the absolute order on
and which are lattices are characterized and some of their
important enumerative invariants are computed.Comment: 26 pages, 6 figures. Theorem 1.3 of the previous version of this
paper is omitted due to a gap in the proof
Towards m-Cambrian Lattices
For positive integers and , we introduce a family of lattices
associated to the Cambrian lattice of
the dihedral group . We show that satisfies
some basic properties of a Fuss-Catalan generalization of ,
namely that and
\bigl\lvert\mathcal{C}_{k}^{(m)}\bigr\rvert=\mbox{Cat}^{(m)}\bigl(I_{2}(k)\bigr).
Subsequently, we prove some structural and topological properties of these
lattices---namely that they are trim and EL-shellable---which were known for
before. Remarkably, our construction coincides in the case
with the -Tamari lattice of parameter 3 due to Bergeron and
Pr{\'e}ville-Ratelle. Eventually, we investigate this construction in the
context of other Coxeter groups, in particular we conjecture that the lattice
completion of the analogous construction for the symmetric group
and the long cycle is isomorphic to the
-Tamari lattice of parameter .Comment: 20 pages, 13 figures. The results of this paper are subsumed by
arXiv:1312.2520, and it will therefore not be publishe
Simply connected homogeneous continua are not separated by arcs
We show that locally connected, simply connected homogeneous continua are not
separated by arcs. We ask several questions about homogeneous continua which
are inspired by analogous questions in geometric group theory.Comment: 15 pages, 8 figure
On the Topology of the Cambrian Semilattices
For an arbitrary Coxeter group , David Speyer and Nathan Reading defined
Cambrian semilattices as semilattice quotients of the weak order
on induced by certain semilattice homomorphisms. In this article, we define
an edge-labeling using the realization of Cambrian semilattices in terms of
-sortable elements, and show that this is an EL-labeling for every
closed interval of . In addition, we use our labeling to show that
every finite open interval in a Cambrian semilattice is either contractible or
spherical, and we characterize the spherical intervals, generalizing a result
by Nathan Reading.Comment: 20 pages, 5 figure
The absolute order on the symmetric group, constructible partially ordered sets and Cohen-Macaulay complexes
The absolute order is a natural partial order on a Coxeter group W. It can be
viewed as an analogue of the weak order on W in which the role of the
generating set of simple reflections in W is played by the set of all
reflections in W. By use of a notion of constructibility for partially ordered
sets, it is proved that the absolute order on the symmetric group is homotopy
Cohen-Macaulay. This answers in part a question raised by V. Reiner and the
first author. The Euler characteristic of the order complex of the proper part
of the absolute order on the symmetric group is also computed.Comment: Final version (only minor changes), 10 pages, one figur
Bijections of dominant regions in the -Shi arrangements of type , and
International audienceIn the present paper, the relation between the dominant regions in the -Shi arrangement of types , and those of the -Shi arrangement of type is investigated. More precisely, it is shown explicitly how the sets and , of dominant regions of the -Shi arrangement of types and respectively, can be projected to the set of dominant regions of the -Shi arrangement of type . This is done by using two different viewpoints for the representative alcoves of these regions: the Shi tableaux and the abacus diagrams. Moreover, bijections between the sets , , and lattice paths inside a rectangle are provided.Dans cet article, nous Ă©tudions la relation entre les rĂ©gions dominantes du -arrangement de Shi de types et ceux du -arrangement de Shi de type . Plus prĂ©cisĂ©ment, nous montrons comment les ensembles et , des rĂ©gions dominantes du -arrangement de Shi de types et respectivement, peuvent ĂȘtre projetĂ©s sur lâensemble des rĂ©gions dominantes du -arrangement de Shi de types . Pour cela nous utilisons deux points de vue diffĂ©rents sur les alcĂŽves reprĂ©sentatives de ces rĂ©gions: les tableaux de Shi et les diagrammes dâabaques. De plus, nous fournissons des bijections entre les ensembles , , et les chemins Ă lâintĂ©rieur dâun rectangle