scholarly journals Climbing elements in finite Coxeter groups

10.37236/428 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Thomas Brady ◽  
Aisling Kenny ◽  
Colum Watt
Keyword(s):  
Coxeter Groups ◽  
Reflection Group ◽  
Total Order ◽  
Coxeter Element

We define the notion of a climbing element in a finite real reflection group relative to a total order on the reflection set and we characterise these elements in the case where the total order arises from a bipartite Coxeter element.

scholarly journals Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Coxeter Groups ◽  
Reflection Group ◽  
Sperner Property ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Noncrossing Partition ◽  
International Audience ◽  
Partition Lattices ◽  
Special Case

International audience We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.

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 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.

scholarly journals Submaximal factorizations of a Coxeter element in complex reflection groups

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Vivien Ripoll
Keyword(s):  
Reflection Group ◽  
Coxeter Element ◽  
Partition Lattice ◽  
Combinatorial Object ◽  
Complex Reflection Groups ◽  
Noncrossing Partition ◽  
Finite Reflection Group ◽  
International Audience ◽  
Cas A ◽  
Noncrossing Partition Lattice

International audience When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $NC(W)$ as a generalized Fuß-Catalan number, depending on the invariant degrees of $W$. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $NC(W)$ as fibers of a "Lyashko-Looijenga covering''. This covering is constructed from the geometry of the discriminant hypersurface of $W$. We deduce new enumeration formulas for certain factorizations of a Coxeter element of $W$. Lorsque $W$ est un groupe de réflexion fini, le treillis $NC(W)$ des partitions non-croisées de type $W$ est un objet combinatoire très riche, qui généralise la notion de partitions non-croisées d'un $n$-gone. Une formule (seulement prouvée au cas par cas à l'heure actuelle) exprime le nombre de chaînes de longueur donnée dans $NC(W)$ sous la forme d'un nombre de Fuß-Catalan généralisé, qui dépend des degrés invariants de $W$. Nous décrivons une stratégie visant à comprendre certaines spécifications de cette formule de manière uniforme, en utilisant une interprétation des chaînes de $NC(W)$ comme fibres d'un "revêtement de Lyashko-Looijenga''. Ce revêtement est construit à partir de la géométrie de l'hypersurface du discriminant de $W$. Nous en déduisons de nouvelles formules de comptage pour certaines factorisations d'un élément de Coxeter de $W$.

Integral Kernels with Reflection Group Invariance

1991 ◽  
Vol 43 (6) ◽  
pp. 1213-1227 ◽  
Author(s):  
Charles F. Dunkl
Keyword(s):  
Coxeter Groups ◽  
Poisson Kernel ◽  
Multivariable Analysis ◽  
Root Systems ◽  
Reflection Group ◽  
Laplacian Operator ◽  
Harmonic Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Group Invariant ◽  
Group Invariant Measure

Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group-invariant measure. The theory has been developed in several papers of the author [4,5,6,7]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.

scholarly journals On non-conjugate Coxeter elements in well-generated reflection groups

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Victor Reiner ◽  
Vivien Ripoll ◽  
Christian Stump
Keyword(s):  
Reflection Group ◽  
Coxeter Element ◽  
Regular Elements ◽  
Coxeter Number ◽  
Complex Reflection Group ◽  
Field Of Definition ◽  
Noncrossing Partition ◽  
Single Orbit ◽  
Nous Montrons ◽  
Common Notion

International audience Given an irreducible well-generated complex reflection group $W$ with Coxeter number $h$, we call a Coxeter element any regular element (in the sense of Springer) of order $h$ in $W$; this is a slight extension of the most common notion of Coxeter element. We show that the class of these Coxeter elements forms a single orbit in $W$ under the action of reflection automorphisms. For Coxeter and Shephard groups, this implies that an element $c$ is a Coxeter element if and only if there exists a simple system $S$ of reflections such that $c$ is the product of the generators in $S$. We moreover deduce multiple further implications of this property. In particular, we obtain that all noncrossing partition lattices of $W$ associated to different Coxeter elements are isomorphic. We also prove that there is a simply transitive action of the Galois group of the field of definition of $W$ on the set of conjugacy classes of Coxeter elements. Finally, we extend several of these properties to Springer's regular elements of arbitrary order. Étant donnés un groupe de réflexion complexe $W$, irréductible et bien engendré, et $h$ son nombre de Coxeter, nous appelons élément de Coxeter un élément régulier (au sens de Springer) d’ordre $h$; ceci est une extension de la notion la plus habituelle d’élément de Coxeter. Nous montrons que l’ensemble de ces éléments de Coxeter forme une seule orbite sous l’action des automorphismes de réflexion de $W$. Pour les groupes de Coxeter et de Shephard, ceci implique qu’un élément $c$ est un élément de Coxeter si et seulement s’il existe un système simple $S$ de réflexions tel que $c$ soit le produit des générateurs dans $S$. Nous déduisons de cette propriété plusieurs autres résultats. En particulier, nous obtenons que tous les treillis de partitions non-croisées de $W$, associés à différents éléments de Coxeter, sont isomorphes. Nous montrons également qu’il existe une action simplement transitive du groupe de Galois du corps de définition de $W$ sur l’ensemble des classes de conjugaison d’éléments de Coxeter. Enfin, nous étendons plusieurs de ces propriétés au cas des éléments réguliers d’ordre quelconque.

scholarly journals Factorization problems in complex reflection groups

2020 ◽  
pp. 1-48
Author(s):  
Joel Brewster Lewis ◽  
Alejandro H. Morales
Keyword(s):  
Space Dimension ◽  
Reflection Group ◽  
Coxeter Element ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Fixed Space Dimension ◽  
Fixed Space

Abstract We enumerate factorizations of a Coxeter element in a well-generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of W-noncrossing partitions.

Synchronicity II

2018 ◽  
Author(s):  
Lexi Eikelboom
Keyword(s):  
Continental Philosophy ◽  
Total Order ◽  
Radical Orthodoxy ◽  
Theological Perspective ◽  
Musical Ontology

In contrast to the previous two chapters, which theologically engage rhythm in continental philosophy, this chapter examines Augustine’s explicitly theological approach to rhythm and its various receptions. The chapter uses Przywara’s scheme of intra-creaturely and theological analogies to frame Augustine’s treatment of rhythm in chapter six of De Musica. While Agamben represents an intra-creaturely perspective, Augustine represents a theological perspective. The degree to which this synchronic, theological view, which envisions rhythm as that which binds metaphysical layers of reality together allowing for communication between them, is problematic depends on the degree to which it is uncoupled from an intra-creaturely perspective like that of Agamben. Proponents of Radical Orthodoxy who propose an Augustinian musical ontology represent such an uncoupling, leading to a total order that betrays creatureliness. Erich Przywara’s interpretation, in contrast, retains the tension in Augustine between both the theological perspective on reality as harmonious and the intra-creaturely experience of interruption.

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