On Counting Types of Symmetries in Finite Unitary Reflection Groups

1979 ◽  
Vol 31 (2) ◽  
pp. 252-254 ◽  
Author(s):  
C. L. Morgan
Keyword(s):  
Reflection Group ◽  
Type I ◽  
Full Group ◽  
Reflection Groups ◽  
Strong Connection ◽  
Linear Automorphism ◽  
Dimensional Vector Space ◽  
Finite Reflection Group ◽  
A Finite Group ◽  
Group Of Symmetries

Let K be a field of characteristic zero. Let V be an n-dimensional vector space over K. A linear automorphism of V is said to be of type i if it leaves fixed a subspace of dimension i. A reflection is a linear automorphism of type n − 1 which has finite order. A finite reflection group is a finite group of linear automorphisms which is generated by reflections. These groups are especially interesting because the full group of symmetries of a regular poly tope is always a finite reflection group. There is also a strong connection between these groups and Lie groups.

scholarly journals Twisted Invariant Theory for Reflection Groups

2006 ◽  
Vol 182 ◽  
pp. 135-170 ◽  
Author(s):  
C. Bonnafé ◽  
G. I. Lehrer ◽  
J. Michel
Keyword(s):  
Invariant Theory ◽  
Regular Element ◽  
Central Element ◽  
Reflection Group ◽  
General Criterion ◽  
Coordinate Ring ◽  
Reflection Groups ◽  
Complex Vector ◽  
Finite Reflection Group ◽  
Module Structures

AbstractLet G be a finite reflection group acting in a complex vector space V = ℂr, whose coordinate ring will be denoted by S. Any element γ ∈ GL(V) which normalises G acts on the ring SG of G-invariants. We attach invariants of the coset Gγ to this action, and show that if G′ is a parabolic subgroup of G, also normalised by γ, the invariants attaching to G′γ are essentially the same as those of Gγ. Four applications are given. First, we give a generalisation of a result of Springer-Stembridge which relates the module structures of the coinvariant algebras of G and G′ and secondly, we give a general criterion for an element of Gγ to be regular (in Springer’s sense) in invariant-theoretic terms, and use it to prove that up to a central element, all reflection cosets contain a regular element. Third, we prove the existence in any well-generated group, of analogues of Coxeter elements of the real reflection groups. Finally, we apply the analysis to quotients of G which are themselves reflection groups.

scholarly journals Discriminants in the invariant theory of reflection groups

1988 ◽  
Vol 109 ◽  
pp. 23-45 ◽  
Author(s):  
Peter Orlik ◽  
Louis Solomon
Keyword(s):  
Invariant Theory ◽  
Reflection Group ◽  
Module Structure ◽  
Polynomial Functions ◽  
Reflection Groups ◽  
Homogeneous Polynomials ◽  
Complex Vector ◽  
Invariant Polynomials ◽  
Basic Set ◽  
Finite Reflection Group

Let V be a complex vector space of dimension l and let G ⊂ GL(V) be a finite reflection group. Let S be the C-algebra of polynomial functions on V with its usual G-module structure (gf)(v) = f{g-1v). Let R be the subalgebra of G-invariant polynomials. By Chevalley’s theorem there exists a set ℬ = {f1, …, fl} of homogeneous polynomials such that R = C[f1, …, fl]. We call ℬ a set of basic invariants or a basic set for G. The degrees di = deg fi are uniquely determined by G. We agree to number them so that d1 ≤ … ≤ di. The map τ: V/G → C1 defined byis a bijection. Each reflection in G fixes some hyperplane in V.

Reflection Subquotients of Unitary Reflection Groups

1999 ◽  
Vol 51 (6) ◽  
pp. 1175-1193 ◽  
Author(s):  
G. I. Lehrer ◽  
T. A. Springer
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Vector Fields ◽  
Reflection Group ◽  
Complex Vector Space ◽  
Reflection Groups ◽  
Complex Vector ◽  
Invariant Vector ◽  
A Finite Group ◽  
Invariant Vector Fields

AbstractLet G be a finite group generated by (pseudo-) reflections in a complex vector space and let g be any linear transformation which normalises G. In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset gG, a subquotient of G which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in G of certain elements of the coset. A criterion is also given in terms of the invariant degrees of G for an integer to be regular for G. A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.

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.

Topological Reflection Groups

1980 ◽  
Vol 32 (2) ◽  
pp. 294-309
Author(s):  
Dragomir Ž. Djoković
Keyword(s):  
Closed Subgroup ◽  
Reflection Group ◽  
Diagonal Matrix ◽  
Analogous Problem ◽  
Compact Groups ◽  
Reflection Groups ◽  
The One

Let G be a closed subgroup of one of the classical compact groups 0(n), U(n), Sp(n). By a reflection we mean a matrix in one of these groups which is conjugate to the diagonal matrix diag (–1, 1, …, 1). We say that G is a topological reflection group (t.r.g.) if the subgroup of G generated by all reflections in G is dense in G.It was shown recently by Eaton and Perlman [5] that, in case of 0(n), the whole group 0(n) is the unique infinite irreducible t.r.g. In this paper we solve the analogous problem for U(n) and Spin). Our method of proof is quite different from the one used in [5]. We treat simultaneously all the three cases.

scholarly journals Gröbner-Shirshov Bases for Temperley-Lieb Algebras of Complex Reflection Groups

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 438
Author(s):  
Jeong-Yup Lee ◽  
Dong-il Lee ◽  
SungSoon Kim
Keyword(s):  
Reflection Group ◽  
Type B ◽  
Reflection Groups ◽  
Combinatorial Interpretation ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Standard Monomials ◽  
Fully Commutative Elements ◽  
The One

We construct a Gröbner-Shirshov basis of the Temperley-Lieb algebra T ( d , n ) of the complex reflection group G ( d , 1 , n ) , inducing the standard monomials expressed by the generators { E i } of T ( d , n ) . This result generalizes the one for the Coxeter group of type B n in the paper by Kim and Lee We also give a combinatorial interpretation of the standard monomials of T ( d , n ) , relating to the fully commutative elements of the complex reflection group G ( d , 1 , n ) . More generally, the Temperley-Lieb algebra T ( d , r , n ) of the complex reflection group G ( d , r , n ) is defined and its dimension is computed.

Harmonic Polynomials Associated With Reflection Groups

2000 ◽  
Vol 43 (4) ◽  
pp. 496-507 ◽  
Author(s):  
Yuan Xu
Keyword(s):  
Weight Function ◽  
Homogeneous Polynomial ◽  
Harmonic Polynomial ◽  
Reflection Group ◽  
Reflection Groups ◽  
Harmonic Polynomials ◽  
Adjoint Operators

AbstractWe extend Maxwell’s representation of harmonic polynomials to h-harmonics associated to a reflection invariant weight function hk. Let 𝑫i, 1 ≤ i ≤ d, be Dunkl’s operators associated with a reflection group. For any homogeneous polynomial P of degree n,we prove the polynomial is a h-harmonic polynomial of degree n, where γ = ∑ki and 𝑫 = (𝑫1, … ,𝑫d). The construction yields a basis for h-harmonics. We also discuss self-adjoint operators acting on the space of h-harmonics.

scholarly journals Distance to the convex hull of an orbit under the action of a compact Lie group

1999 ◽  
Vol 66 (3) ◽  
pp. 331-357 ◽  
Author(s):  
Randall R. Holmes ◽  
Tin-Yau Tam
Keyword(s):  
Convex Hull ◽  
Analogous Result ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Adjoint Action ◽  
Reflection Group ◽  
Compact Lie Group ◽  
Real Vector ◽  
Finite Reflection Group

AbstractFor a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

scholarly journals Invariants of Finite Reflection Groups

1960 ◽  
Vol 12 ◽  
pp. 616-618 ◽  
Author(s):  
Robert Steinberg
Keyword(s):  
Unitary Transformation ◽  
Reflection Group ◽  
Polynomial Basis ◽  
Reflection Groups ◽  
Polynomial Invariants

Let us define a reflection to be a unitary transformation, other than the identity, which leaves fixed, pointwise, a (reflecting) hyperplane, that is, a subspace of deficiency 1, and a reflection group to be a group generated by reflections. Chevalley (1) (and also Coxeter (2) together with Shephard and Todd (4)) has shown that a reflection group G, acting on a space of n dimensions, possesses a set of n algebraically independent (polynomial) invariants which form a polynomial basis for the set of all invariants of G.

SHAPIRO’S UNCERTAINTY PRINCIPLE IN THE DUNKL SETTING

2015 ◽  
Vol 92 (1) ◽  
pp. 98-110 ◽  
Author(s):  
SAIFALLAH GHOBBER
Keyword(s):  
Uncertainty Principle ◽  
Orthonormal Basis ◽  
Integral Transform ◽  
Reflection Group ◽  
Dunkl Transform ◽  
Time Frequency ◽  
Orthonormal Bases ◽  
Orthonormal Sequence ◽  
Finite Reflection Group ◽  
Uniformly Bounded

The Dunkl transform ${\mathcal{F}}_{k}$ is a generalisation of the usual Fourier transform to an integral transform invariant under a finite reflection group. The goal of this paper is to prove a strong uncertainty principle for orthonormal bases in the Dunkl setting which states that the product of generalised dispersions cannot be bounded for an orthonormal basis. Moreover, we obtain a quantitative version of Shapiro’s uncertainty principle on the time–frequency concentration of orthonormal sequences and show, in particular, that if the elements of an orthonormal sequence and their Dunkl transforms have uniformly bounded dispersions then the sequence is finite.

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