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.

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 The Hessian map in the invariant theory of reflection groups

1988 ◽  
Vol 109 ◽  
pp. 1-21 ◽  
Author(s):  
Peter Orlik ◽  
Louis Solomon
Keyword(s):  
Vector Space ◽  
Invariant Theory ◽  
Polynomial Functions ◽  
Complex Vector Space ◽  
Dual Basis ◽  
Reflection Groups ◽  
Complex Vector ◽  
Module Homomorphism

Let V be a complex vector space of dimension l. Let S be the C-algebra of polynomial functions on V. Let Ders be the S-module of derivations of S and let Ωs = Homs (Ders, S) be the dual S-module of differential 1-forms. Let {ei} be a basis for V and let {xi} be the dual basis for V. Then {Di = ∂/∂xi and {dxi} are bases for Ders and Ωs as S-modules. If f ∈ S, define a map Hess (f): Ders → Ωs byThen Hess (f) is an S-module homomorphism which does not depend on the choice of basis for V.

scholarly journals Note on Cubature Formulae and Designs Obtained from Group Orbits

2012 ◽  
Vol 64 (6) ◽  
pp. 1359-1377 ◽  
Author(s):  
Hiroshi Nozaki ◽  
Masanori Sawa
Keyword(s):  
Cubature Formula ◽  
Sufficient Conditions ◽  
Reflection Group ◽  
Alternative Proof ◽  
Invariant Polynomials ◽  
Cubature Formulas ◽  
Finite Reflection Group ◽  
Necessary And Sufficient ◽  
Cubature Formulae ◽  
Invariant Cubature Formulas

Abstract In 1960, Sobolev proved that for a finite reflection group G, a G-invariant cubature formula is of degree t if and only if it is exact for all G-invariant polynomials of degree at most t . In this paper, we make some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and, moreover, gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007), which classifies tight Euclidean designs invariant under the Weyl group of type B, to other finite reflection groups.

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 Chevalley's theorem in class Cr

2009 ◽  
Vol 139 (4) ◽  
pp. 743-758
Author(s):  
Gérard P. Barbançon
Keyword(s):  
Positive Integer ◽  
Reflection Group ◽  
Linear Mapping ◽  
Polynomial Mapping ◽  
Integer Part ◽  
Invariant Polynomials ◽  
Linear Forms ◽  
Counter Example ◽  
Finite Reflection Group

Let W be a finite reflection group acting orthogonally on ℝn, P be the Chevalley polynomial mapping determined by an integrity basis of the algebra of W-invariant polynomials, and h be the highest degree of the coordinate polynomials in P. Let r be a positive integer and [r/h] be the integer part of r/h. There exists a linear mapping $\mathcal{C}^r(\mathbb{R}^n)^W\ni f\mapsto F\in\mathcal{C}^{[r/h]}(\mathbb{R}^n)$ such that f = F ∘ P, which is continuous for the natural Fréchet topologies. A general counter-example shows that this result is the best possible. The proof uses techniques of division by linear forms and a study of compensation phenomena. An extension to P−1(ℝn) of invariant formally holomorphic regular fields is needed.

scholarly journals Differential operators and reection group of type Bn

2021 ◽  
Vol 2090 (1) ◽  
pp. 012097
Author(s):  
Ibrahim Nonkané ◽  
M. Latévi Lawson
Keyword(s):  
Polynomial Ring ◽  
Weyl Algebra ◽  
Differential Operators ◽  
Reflection Group ◽  
Type B ◽  
Invariant Polynomials ◽  
Polynomial Representation ◽  
Invariant Differential Operators ◽  
Finite Reflection Group ◽  
Invariant Differential

Abstract In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group W of type Bn. We endowed the polynomial ring C[x 1,..., xn ] with a structure of module over the Weyl algebra associated with the ring C[x 1,..., xn]W of invariant polynomials under a reflections group W of type Bn . Then we study the polynomials representation of the ring of invariant differential operators under the reflections group W. We make use of the theory of representation of groups namely the higher Specht polynomials associated with the reflection group W to yield a decomposition of that structure by providing explicitly the generators of its simple components.

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.

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