Classification of Coxeter systems and reflection groups

2001 ◽  
pp. 81-96
Author(s):  
Richard Kane ◽  
Jonathan Borwein ◽  
Peter Borwein
Keyword(s):  
Reflection Groups ◽  
Coxeter Systems

scholarly journals A Refined Count of Coxeter Element Reflection Factorizations

10.37236/7362 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Elise DelMas ◽  
Thomas Hameister ◽  
Victor Reiner
Keyword(s):  
Generating Function ◽  
Coxeter Element ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Coxeter Number ◽  
Simple Product

For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the numberof reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.

Voronoi Domains and Dual Cells in the Generalized Kaleidoscope with Applications to Root and Weight Lattices

1995 ◽  
Vol 47 (3) ◽  
pp. 573-605 ◽  
Author(s):  
R. V. Moody ◽  
J. Patera
Keyword(s):  
General Theory ◽  
Lie Algebra ◽  
Reflection Groups ◽  
Semisimple Lie Algebra ◽  
Voronoi Cells ◽  
Special Cases

AbstractWe give a uniform description, in terms of Coxeter diagrams, of the Voronoi domains of the root and weight lattices of any semisimple Lie algebra. This description provides a classification not only of all the facets of these Voronoi domains but simultaneously a classification of their dual or Delaunay cells and their facets. It is based on a much more general theory that we develop here providing the same sort of information in the setting of chamber geometries defined by arbitrary reflection groups. These generalized kaleidoscopes include the classical spherical, Euclidean, and hyperbolic kaleidoscopes as special cases. We prove that under certain conditions the Delaunay cells are Voronoi cells for the vertices of the Voronoi complex. This leads to the description in terms of Wythoff polytopes of the Voronoi cells of the weight lattices.

scholarly journals $\vee$ -Systems, Holonomy Lie Algebras, and Logarithmic Vector Fields

2017 ◽  
Vol 2018 (7) ◽  
pp. 2070-2098 ◽  
Author(s):  
Misha V Feigin ◽  
Alexander P Veselov
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Special Class ◽  
Vector Fields ◽  
Hyperplane Arrangement ◽  
Gradient Vector ◽  
Coxeter Systems ◽  
Flat Coordinates

Abstract It is shown that the description of certain class of representations of the holonomy Lie algebra $\mathfrak g_{\Delta}$ associated with hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$-systems associated with $\Delta.$ The flat sections of the corresponding $\vee$-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any $\vee$-system is free in Saito's sense and show this for all known $\vee$-systems and for a special class of $\vee$-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic $\vee$-systems.

CLASSIFICATION OF REFLECTION SUBGROUPS MINIMALLY CONTAINING -SYLOW SUBGROUPS

2017 ◽  
Vol 97 (1) ◽  
pp. 57-68 ◽  
Author(s):  
KANE DOUGLAS TOWNSEND
Keyword(s):  
Conjugacy Class ◽  
Prime Divisor ◽  
Sylow Subgroup ◽  
Reflection Group ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Sylow Subgroups

Let a prime $p$ divide the order of a finite real reflection group. We classify the reflection subgroups up to conjugacy that are minimal with respect to inclusion, subject to containing a $p$-Sylow subgroup. For Weyl groups, this is achieved by an algorithm inspired by the Borel–de Siebenthal algorithm. The cases where there is not a unique conjugacy class of reflection subgroups minimally containing the $p$-Sylow subgroups are the groups of type $F_{4}$ when $p=2$ and $I_{2}(m)$ when $m\geq 6$ is even but not a power of $2$ for each odd prime divisor $p$ of $m$. The classification significantly reduces the cases required to describe the $p$-Sylow subgroups of finite real reflection groups.

Reflection groups and Coxeter systems

2001 ◽  
pp. 66-74
Author(s):  
Richard Kane ◽  
Jonathan Borwein ◽  
Peter Borwein
Keyword(s):  
Reflection Groups ◽  
Coxeter Systems
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