Classification of finite reflection groups

1990 ◽  
pp. 29-48
Keyword(s):  
Reflection Groups

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.

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.

scholarly journals Canonical Systems of Basic Invariants for Unitary Reflection Groups

2016 ◽  
Vol 59 (3) ◽  
pp. 617-623 ◽  
Author(s):  
Norihiro Nakashima ◽  
Hiroaki Terao ◽  
Shuhei Tsujie
Keyword(s):  
Explicit Formula ◽  
Reflection Group ◽  
Canonical System ◽  
Reflection Groups ◽  
Canonical Systems ◽  
The Third ◽  
Unitary Reflection ◽  
Unitary Reflection Group

AbstractIt is known that there exists a canonical system for every finite real reflection group. In a previous paper, the first and the third authors obtained an explicit formula for a canonical system. In this article, we first define canonical systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a canonical system for every unitary reflection group.

scholarly journals Classification of Vertex-Transitive Zonotopes

2021 ◽  
Author(s):  
Martin Winter
Keyword(s):  
Root Systems ◽  
Reflection Group ◽  
Sufficient Condition ◽  
Reflection Groups ◽  
Centrally Symmetric ◽  
Finite Reflection Group ◽  
Vertex Transitive ◽  
Symmetric Set ◽  
Full Classification

AbstractWe give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a $$\Gamma $$ Γ -permutahedron for some finite reflection group $$\Gamma \subset {{\,\mathrm{O}\,}}(\mathbb {R}^d)$$ Γ ⊂ O ( R d ) . The same holds true for zonotopes in which all vertices are on a common sphere, and all edges are of the same length. The classification of these then follows from the classification of finite reflection groups. We prove that root systems can be characterized as those centrally symmetric sets of vectors, for which all intersections with half-spaces, that contain exactly half the vectors, are congruent. We provide a further sufficient condition for a centrally symmetric set to be a root system.

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