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 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 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 The Minimal Growth Rate of Cocompact Coxeter Groups in Hyperbolic 3-space

2014 ◽  
Vol 66 (2) ◽  
pp. 354-372 ◽  
Author(s):  
Ruth Kellerhals ◽  
Alexander Kolpakov
Keyword(s):  
Growth Rate ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Triangle Group ◽  
Minimal Growth ◽  
Salem Number

AbstractDue to work of W. Parry it is known that the growth rate of a hyperbolic Coxeter group acting cocompactly on H3 is a Salem number. This being the arithmetic situation, we prove that the simplex group (3,5,3) has the smallest growth rate among all cocompact hyperbolic Coxeter groups, and that it is, as such, unique. Our approach provides a different proof for the analog situation in H2 where E. Hironaka identified Lehmer's number as the minimal growth rate among all cocompact planar hyperbolic Coxeter groups and showed that it is (uniquely) achieved by the Coxeter triangle group (3,7).

scholarly journals Chiral Polyhedra Derived from Coxeter Diagrams and Quaternions

2011 ◽  
Vol 16 ◽  
pp. 82 ◽  
Author(s):  
Mehmet Koca ◽  
Nazife Ozdes Koca ◽  
Muna Al-Shueili
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Linear Combinations ◽  
Octahedral Group ◽  
Snub Cube ◽  
Proper Rotation

There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. In this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Coxeter groups and to derive the orbits representing the solids of interest. They lead to the polyhedra tetrahedron, icosahedron, snub cube, and snub dodecahedron respectively. We prove that the tetrahedron and icosahedron can be transformed to their mirror images by the proper rotational octahedral group so they are not classified in the class of chiral polyhedra. It is noted that vertices of the snub cube and snub dodecahedron can be derived from the vectors, which are linear combinations of the simple roots, by the actions of the proper rotation groupsand  respectively. Their duals are constructed as the unions of three orbits of the groups of concern. We also construct the polyhedra, quasiregular in general, by combining chiral polyhedra with their mirror images. As a by-product we obtain the pyritohedral group as the subgroup the Coxeter group and discuss the constructions of pyritohedrons. We employ a method which describes the Coxeter groups and their orbits in terms of quaternions.  

On the generation of Coxeter groups and their alternating subgroups by involutions

2019 ◽  
Vol 22 (6) ◽  
pp. 1001-1013
Author(s):  
Rupert W. T. Yu
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Minimum Number

Abstract In this note, we determine the minimum number of involutions required to generate a finite irreducible Coxeter group, and also whenever such generation is possible, its alternating subgroup. Explicit generators are given.

scholarly journals Weak Generalized Lifting Property, Bruhat Intervals, and Coxeter Matroids

2020 ◽  
Author(s):  
Fabrizio Caselli ◽  
Michele D’Adderio ◽  
Mario Marietti
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Combinatorial Properties ◽  
Coxeter Matroid ◽  
Finite Coxeter Group ◽  
Bruhat Intervals

Abstract We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.

scholarly journals Gröbner–Shirshov bases for non-crystallographic Coxeter groups

2019 ◽  
Vol 75 (3) ◽  
pp. 584-592 ◽  
Author(s):  
Jeong-Yup Lee ◽  
Dong-il Lee
Keyword(s):  
Group Algebra ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Standard Monomials ◽  
Natural Operation

For the group algebra of the finite non-crystallographic Coxeter group of type H 4, its Gröbner–Shirshov basis is constructed as well as the corresponding standard monomials, which describe explicitly all symmetries acting on the 120-cell and produce a natural operation table between the 14400 elements for the group.

scholarly journals On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings

2009 ◽  
Vol 61 (4) ◽  
pp. 740-761 ◽  
Author(s):  
Pierre-Emmanuel Caprace ◽  
Frédéric Haglund
Keyword(s):  
Abelian Group ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Free Abelian Group ◽  
Gromov Hyperbolic ◽  
Convex Cocompact ◽  
Rank 2 ◽  
Cocompact Subgroup ◽  
Special Case ◽  
Geometric Action

Abstract.Given a complete CAT(0) space X endowed with a geometric action of a group Ⲅ, it is known that if Ⲅ contains a free abelian group of rank n, then X contains a geometric flat of dimension n. We prove the converse of this statement in the special case where X is a convex subcomplex of the CAT(0) realization of a Coxeter group W, and Ⲅ is a subgroup of W. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings.

scholarly journals Conjugacy classes of involutions in Coxeter groups

1982 ◽  
Vol 26 (1) ◽  
pp. 1-15 ◽  
Author(s):  
R.W. Richardson
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Equivalence Classes ◽  
Conjugacy Classes ◽  
Elementary Method

In this paper we give an elementary method for classifying conjugacy classes of involutions in a Coxeter group (W, S). The classification is in terms of (W-equivalence classes of certain subsets of S).

scholarly journals Mixed Perverse Sheaves on Flag Varieties for Coxeter Groups

2019 ◽  
Vol 72 (1) ◽  
pp. 1-55
Author(s):  
Pramod N. Achar ◽  
Simon Riche ◽  
Cristian Vay
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Grothendieck Group ◽  
General Setting ◽  
Hecke Algebra ◽  
Abelian Category ◽  
Flag Varieties ◽  
Perverse Sheaves

AbstractIn this paper we construct an abelian category of mixed perverse sheaves attached to any realization of a Coxeter group, in terms of the associated Elias–Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias–Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.

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