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 On the centralizer algebra of the unitary reflection group G(m,p,n)

1997 ◽  
Vol 148 ◽  
pp. 113-126 ◽  
Author(s):  
Kenichiro Tanabe
Keyword(s):  
Tensor Product ◽  
Vector Space ◽  
Symmetric Group ◽  
Reflection Group ◽  
Unitary Reflection ◽  
Unitary Reflection Group ◽  
Group Case ◽  
Centralizer Algebra

AbstractThe imprimitive unitary reflection group G(m, p, n) acts on the vector space V =Cn naturally. The symmetric group Sk acts on ⊗kV by permuting the tensor product factors. We show that the algebra of all matrices on ⊗kV commuting with G(m, p, n) is generated by Sk and three other elements. This is a generalization of Jones’s results for the symmetric group case [J].

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.

CHARACTERIZATION FOR THE MODULAR PARTY ALGEBRA

2008 ◽  
Vol 17 (08) ◽  
pp. 939-960 ◽  
Author(s):  
M. KOSUDA
Keyword(s):  
Positive Integer ◽  
Endomorphism Ring ◽  
Reflection Group ◽  
Generators And Relations ◽  
Unitary Reflection ◽  
Unitary Reflection Group

In this paper, we give a characterization for the modular party algebra Pn,r(Q) by generators and relations. By specializing the parameter Q to a positive integer k, this algebra becomes the centralizer of the unitary reflection group G(r, 1, k) in the endomorphism ring of V⊗n under the condition that k ≥ n.

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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