Relative angular measure of the dual cone of the fundamental cone of a finite reflection group

2010 ◽  
Vol 65 (1) ◽  
pp. 173-174
Author(s):  
Pavel V Bibikov
Keyword(s):  
Reflection Group ◽  
Dual Cone ◽  
Angular Measure ◽  
Finite Reflection Group ◽  
Fundamental Cone

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 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.

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.

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 Negative Powers of Laguerre Operators

2012 ◽  
Vol 64 (1) ◽  
pp. 183-216 ◽  
Author(s):  
Adam Nowak ◽  
Krzysztof Stempak
Keyword(s):  
Harmonic Oscillator ◽  
Differential Operators ◽  
Reflection Group ◽  
Convolution Type ◽  
Laguerre Function ◽  
Kernel Estimates ◽  
Hermite Operator ◽  
Finite Reflection Group ◽  
Straightforward Approach ◽  
Continuous Range

Abstract We study negative powers of Laguerre differential operators in ℝd, d ≥ 1. For these operators we prove two-weight Lp − Lq estimates with ranges of q depending on p. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices. we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the convexity principle to cover the continuous range of ϵ 2 [−1/2, ∞)d. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on ℝd and isomorphic to ℤd2. The two weight Lp − Lq estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type.

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 k-Parabolic Subspace Arrangements

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Christopher Severs ◽  
Jacob White
Keyword(s):  
Fundamental Group ◽  
Reflection Group ◽  
Artin Group ◽  
Subspace Arrangements ◽  
Coxeter Arrangement ◽  
Algebraic Description ◽  
Finite Reflection Group ◽  
International Audience

International audience In this paper, we study k-parabolic arrangements, a generalization of the k-equal arrangement for any finite real reflection group. When k=2, these arrangements correspond to the well-studied Coxeter arrangements. Brieskorn (1971) showed that the fundamental group of the complement of the type W Coxeter arrangement (over $\mathbb{C}$) is isomorphic to the pure Artin group of type W. Khovanov (1996) gave an algebraic description for the fundamental group of the complement of the 3-equal arrangement (over $\mathbb{R}$). We generalize Khovanov's result to obtain an algebraic description of the fundamental group of the complement of the 3-parabolic arrangement for arbitrary finite reflection group. Our description is a real analogue to Brieskorn's description. Nous généralisons les arrangements k-égaux à tous les groupes de réflexions finis réels. Les arrangements ainsi obtenus sont dits k-paraboliques. Dans le cas où k = 2 nous retrouvons les arrangements de Coxeter qui sont bien connus. En 1971, Brieskorn démontra que le groupe fondamental associé au complément (complexe) de l'arrangement de Coxeter de type W est en fait isomorphe au groupe pure d'Artin de type W . En 1996, Khovanov donne une description algébrique du groupe fondamental du complément (réel) de l'arrangement 3-égaux. Nous généralisons le résultat de Khovanov et obtenons une description algébrique du groupe fondamental de l'espace complément d'un arrangement k-parabolique pour tous les groupes de réflexions finis et réels. Il se trouve que notre description est l'analogue réel de la description de Brieskorn.

On the boundedness of the Dunkl spherical maximal operator

2016 ◽  
Vol 08 (03) ◽  
pp. 475-495 ◽  
Author(s):  
Luc Deleaval
Keyword(s):  
Maximal Operator ◽  
Reflection Group ◽  
Square Functions ◽  
Finite Reflection Group ◽  
Multiplier Operators

In this paper, we introduce a Dunkl-type spherical maximal operator associated with a finite reflection group and study its boundedness. In view of this study, we give some results on suitable square functions and maximal multiplier operators in the Dunkl setting.

On a certain generator system of the ring of invariants of a finite reflection group

1980 ◽  
Vol 8 (4) ◽  
pp. 373-408 ◽  
Author(s):  
Kyoji Saito ◽  
Tamaki Yano ◽  
Jiro Sekiguchi
Keyword(s):  
Reflection Group ◽  
Generator System ◽  
Finite Reflection Group
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