A Combinatorial Construction of the Weak Order of a Coxeter Group

For an arbitrary Coxeter group $W$, Reading and Speyer defined Cambrian semilattices $\mathcal{C}_{\gamma}$ as sub-semilattices of the weak order on $W$ induced by so-called $\gamma$-sortable elements. In this article, we define an edge-labeling of $\mathcal{C}_{\gamma}$, and show that this is an EL-labeling for every closed interval of $\mathcal{C}_{\gamma}$. 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 Reading.

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Affine Permutations of Type $A$

The Electronic Journal of Combinatorics ◽

10.37236/1276 ◽

1995 ◽

Vol 3 (2) ◽

Cited By ~ 15

Author(s):

Anders Björner ◽

Francesco Brenti

Keyword(s):

Coxeter Group ◽

Type A ◽

Bruhat Order ◽

Weak Order ◽

Combinatorial Properties ◽

Affine Permutations

We study combinatorial properties, such as inversion table, weak order and Bruhat order, for certain infinite permutations that realize the affine Coxeter group $\tilde{A}_{n}$.

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The facial weak order in finite Coxeter groups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6399 ◽

2020 ◽

Vol DMTCS Proceedings, 28th... ◽

Author(s):

Aram Dermenjian ◽

Christophe Hohlweg ◽

Vincent Pilaud

Keyword(s):

Coxeter Group ◽

Coxeter Groups ◽

Geometric Interpretation ◽

Weak Order ◽

Maximal Length ◽

Lattice Congruence ◽

Finite Coxeter Group ◽

International Audience

International audience We investigate a poset structure that extends the weak order on a finite Coxeter group W to the set of all faces of the permutahedron of W. We call this order the facial weak order. We first provide two alternative characterizations of this poset: a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial, that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Bjo ̈rner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of its classes.

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The Weak Order on Weyl Posets

Canadian Journal of Mathematics ◽

10.4153/s0008414x19000063 ◽

2019 ◽

Vol 72 (4) ◽

pp. 867-899

Author(s):

Joël Gay ◽

Vincent Pilaud

Keyword(s):

Weyl Group ◽

Root System ◽

Coxeter Group ◽

Lattice Structure ◽

Root Systems ◽

Weak Order ◽

Generalized Associahedra

AbstractWe define a natural lattice structure on all subsets of a finite root system that extends the weak order on the elements of the corresponding Coxeter group. For crystallographic root systems, we show that the subposet of this lattice induced by antisymmetric closed subsets of roots is again a lattice. We then study further subposets of this lattice that naturally correspond to the elements, the intervals, and the faces of the permutahedron and the generalized associahedra of the corresponding Weyl group. These results extend to arbitrary finite crystallographic root systems the recent results of G. Chatel, V. Pilaud, and V. Pons on the weak order on posets and its induced subposets.

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Sortable Elements for Quivers with Cycles

The Electronic Journal of Combinatorics ◽

10.37236/362 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 2

Author(s):

Nathan Reading ◽

David E Speyer

Keyword(s):

Coxeter Group ◽

Weak Order ◽

Cluster Algebras ◽

Combinatorial Property ◽

General Notion ◽

Arbitrary Orientation ◽

Coxeter Element ◽

Acyclic Orientation ◽

Coxeter Diagram ◽

Carry Over

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.

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On the Topology of the Cambrian Semilattices

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2320 ◽

2013 ◽

Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽

Author(s):

Myrto Kallipoliti ◽

Henri Mühle

Keyword(s):

Coxeter Group ◽

Closed Interval ◽

Open Interval ◽

Weak Order ◽

International Audience ◽

Nous Utilisons

International audience For an arbitrary Coxeter group $W$, David Speyer and Nathan Reading defined Cambrian semilattices $C_{\gamma}$ as certain sub-semilattices of the weak order on $W$. In this article, we define an edge-labelling using the realization of Cambrian semilattices in terms of $\gamma$-sortable elements, and show that this is an EL-labelling for every closed interval of $C_{\gamma}$. In addition, we use our labelling 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. Pour tout groupe de Coxeter $W$, David Speyer et Nathan Reading ont défini les demi-treillis Cambriens comme certains sous-demi-treillis de l’ordre faible sur $W$. Dans cet article, nous définissons un étiquetage des arêtes basé sur la réalisation des demi-treillis Cambriens en termes d’éléments$\gamma$-triables, et prouvons que c’est un étiquetage EL pour tout intervalle fermé de $C_{\gamma}$. Nous utilisons de plus cet étiquetage pour ontrer que tout intervalle ouvert fini dans un demi-treillis Cambrien est soit contractile soit sphérique, et nous caractérisons les intervalles sphériques,généralisant ainsi un résultat de Nathan Reading.

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Tits Indices

10.23943/princeton/9780691166902.003.0020 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Coxeter Group ◽

Coxeter System ◽

Relative Rank ◽

Coxeter Diagram ◽

The Absolute ◽

Relative Type

This chapter introduces the notion of a Tits index and the notion of the relative Coxeter diagram of a Tits index. It first defines a Tits index, which can be anisotropic or isotropic, quasi-split or split, before considering a number of propositions regarding compatible representations. It then gives a proof of the theorem that includes two assumptions about a Coxeter system, focusing on the absolute Coxeter system, the relative Coxeter system, and the relative Coxeter group of the Tits index, as well as the absolute Coxeter diagram (or absolute type), the relative Coxeter diagram (or relative type), and the absolute rank and the relative rank of the Tits index. The chapter concludes with some observations about the case that (W, S) is spherical, irreducible or affine.

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