scholarly journals Sortable Elements for Quivers with Cycles

10.37236/362 ◽  
2010 ◽  
Vol 17 (1) ◽  
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.

scholarly journals SB-Labelings, Distributivity, and Bruhat Order on Sortable Elements

10.37236/4942 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Bruhat Order ◽  
Distributive Lattices ◽  
Coxeter Element ◽  
General Statement ◽  
Coxeter Diagram

In this article, we investigate the set of $\gamma$-sortable elements, associated with a Coxeter group $W$ and a Coxeter element $\gamma\in W$, under Bruhat order, and we denote this poset by $\mathcal{B}_{\gamma}$. We show that this poset belongs to the class of SB-lattices recently introduced by Hersh and Mészáros, by proving a more general statement, namely that all join-distributive lattices are SB-lattices. The observation that $\mathcal{B}_{\gamma}$ is join-distributive is due to Armstrong. Subsequently, we investigate for which finite Coxeter groups $W$ and which Coxeter elements $\gamma\in W$ the lattice $\mathcal{B}_{\gamma}$ is in fact distributive. It turns out that this is the case for the "coincidental" Coxeter groups, namely the groups $A_{n},B_{n},H_{3}$ and $I_{2}(k)$. We conclude this article with a conjectural characteriziation of the Coxeter elements $\gamma$ of said groups for which $\mathcal{B}_{\gamma}$ is distributive in terms of forbidden orientations of the Coxeter diagram.

Tits Indices

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.

A Combinatorial Construction of the Weak Order of a Coxeter Group

2005 ◽  
Vol 33 (5) ◽  
pp. 1447-1460 ◽  
Author(s):  
Přemysl Jedlička
Keyword(s):  
Coxeter Group ◽  
Weak Order ◽  
Combinatorial Construction

scholarly journals On the Topology of the Cambrian Semilattices

10.37236/2910 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Myrto Kallipoliti ◽  
Henri Mühle
Keyword(s):  
Coxeter Group ◽  
Closed Interval ◽  
Open Interval ◽  
Weak Order ◽  
Edge Labeling

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.

scholarly journals Affine Permutations of Type $A$

10.37236/1276 ◽  
1995 ◽  
Vol 3 (2) ◽  
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 $\tilde{A}_{n}$.

scholarly journals On the Hurwitz action in finite Coxeter groups

2017 ◽  
Vol 20 (1) ◽  
Author(s):  
Barbara Baumeister ◽  
Thomas Gobet ◽  
Kieran Roberts ◽  
Patrick Wegener
Keyword(s):  
Parabolic Subgroup ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Coxeter Element ◽  
Necessary And Sufficient Condition ◽  
Reduced Decompositions ◽  
Finite Coxeter Group ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Hurwitz Action

AbstractWe provide a necessary and sufficient condition on an element of a finite Coxeter group to ensure the transitivity of the Hurwitz action on its set of reduced decompositions into products of reflections. We show that this action is transitive if and only if the element is a parabolic quasi-Coxeter element. We call an element of the Coxeter group parabolic quasi-Coxeter element if it has a factorization into a product of reflections that generate a parabolic subgroup. We give an unusual definition of a parabolic subgroup that we show to be equivalent to the classical one for finite Coxeter groups.

scholarly journals Conjugacy of Coxeter Elements

10.37236/70 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Henrik Eriksson ◽  
Kimmo Eriksson
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Group Element ◽  
Coxeter Element ◽  
Special Cases

For a Coxeter group $(W,S)$, a permutation of the set $S$ is called a Coxeter word and the group element represented by the product is called a Coxeter element. Moving the first letter to the end of the word is called a rotation and two Coxeter elements are rotation equivalent if their words can be transformed into each other through a sequence of rotations and legal commutations. We prove that Coxeter elements are conjugate if and only if they are rotation equivalent. This was known for some special cases but not for Coxeter groups in general.

scholarly journals The facial weak order in finite Coxeter groups

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.

scholarly journals The Weak Order on Weyl Posets

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.

scholarly journals Autocatalytic networks in biology: structural theory and algorithms

2019 ◽  
Vol 16 (151) ◽  
pp. 20180808 ◽  
Author(s):  
Mike Steel ◽  
Wim Hordijk ◽  
Joana C. Xavier
Keyword(s):  
Mathematical Analysis ◽  
General Setting ◽  
Structural Theory ◽  
Living Systems ◽  
General Notion ◽  
Graph Theoretic ◽  
Polynomial Time Algorithms ◽  
The Origin Of Life ◽  
Carry Over ◽  
Autocatalytic Networks

Self-sustaining autocatalytic networks play a central role in living systems, from metabolism at the origin of life, simple RNA networks and the modern cell, to ecology and cognition. A collectively autocatalytic network that can be sustained from an ambient food set is also referred to more formally as a ‘reflexively autocatalytic food-generated’ (RAF) set. In this paper, we first investigate a simplified setting for studying RAFs, which is nevertheless relevant to real biochemistry and which allows an exact mathematical analysis based on graph-theoretic concepts. This, in turn, allows for the development of efficient (polynomial-time) algorithms for questions that are computationally intractable (NP-hard) in the general RAF setting. We then show how this simplified setting for RAF systems leads naturally to a more general notion of RAFs that are ‘generative’ (they can be built up from simpler RAFs) and for which efficient algorithms carry over to this more general setting. Finally, we show how classical RAF theory can be extended to deal with ensembles of catalysts as well as the assignment of rates to reactions according to which catalysts (or combinations of catalysts) are available.

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