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 Deformed diagonal harmonic polynomials for complex reflection groups

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
François Bergeron ◽  
Nicolas Borie ◽  
Nicolas M. Thiéry
Keyword(s):  
Reflection Group ◽  
Supporting Evidence ◽  
Finite Complex ◽  
Reflection Groups ◽  
Harmonic Polynomials ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
International Audience ◽  
Diagonal Harmonic

arXiv : http://arxiv.org/abs/1011.3654 International audience We introduce deformations of the space of (multi-diagonal) harmonic polynomials for any finite complex reflection group of the form W=G(m,p,n), and give supporting evidence that this space seems to always be isomorphic, as a graded W-module, to the undeformed version. Nous introduisons une déformation de l'espace des polynômes harmoniques (multi-diagonaux) pour tout groupe de réflexions complexes de la forme W=G(m,p,n), et soutenons l'hypothèse que cet espace est toujours isomorphe, en tant que W-module gradué, à l'espace d'origine.

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 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 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.

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.

scholarly journals Hospital ethics reflection groups: a learning and development resource for clinical practice

2019 ◽  
Vol 20 (1) ◽  
Author(s):  
H. Bruun ◽  
L. Huniche ◽  
E. Stenager ◽  
C. B. Mogensen ◽  
R. Pedersen
Keyword(s):  
Clinical Practice ◽  
General Hospital ◽  
Positive Impact ◽  
Reflection Group ◽  
Ethical Challenge ◽  
Ethical Challenges ◽  
Reflection Groups ◽  
Patient Admissions ◽  
The Impact ◽  
Ethics Reflection

Abstract Background An ethics reflection group (ERG) is one of a number of ethics support services developed to better handle ethical challenges in healthcare. The aim of this article is to evaluate the significance of ERGs in psychiatric and general hospital departments in Denmark. Methods This is a qualitative action research study, including systematic text condensation of 28 individual interviews and 4 focus groups with clinicians, ethics facilitators and ward managers. Short written descriptions of the ethical challenges presented in the ERGs also informed the analysis of significance. Results A recurring ethical challenge for clinicians, in a total of 63 cases described and assessed in 3 ethical reflection groups, is to strike a balance between respect for patient autonomy, paternalistic responsibility, professional responsibilities and institutional values. Both in psychiatric and general hospital departments, the study participants report a positive impact of ERG, which can be divided into three categories: 1) Significance for patients, 2) Significance for clinicians, and 3) Significance for ward managers. In wards characterized by short-time patient admissions, the cases assessed were retrospective and the beneficiaries of improved dialogue mainly future patients rather than the patients discussed in the specific ethical challenge presented. In wards with longer admissions, the patients concerned also benefitted from the dialogue in the ERG. Conclusion This study indicates a positive significance and impact of ERGs; constituting an interdisciplinary learning resource for clinicians, creating significance for themselves, the ward managers and the organization. By introducing specific examples, this study indicates that ERGs have significance for the patients discussed in the specific ethical challenge, but mostly indirectly through learning among clinicians and development of clinical practice. More research is needed to further investigate the impact of ERGs seen from the perspectives of patients and relatives.

scholarly journals Experiences of moral challenges in everyday nursing practice: In light of healthcare professionals’ self-understanding

2016 ◽  
Vol 36 (4) ◽  
pp. 177-183 ◽  
Author(s):  
Margareth Kristoffersen ◽  
Febe Friberg ◽  
Berit Støre Brinchmann
Keyword(s):  
Nursing Practice ◽  
Healthcare Professionals ◽  
Reflection Group ◽  
Reflection Groups ◽  
Comprehensive Understanding ◽  
Organizational Demands ◽  
Hermeneutic Approach

This study has two aims: firstly, to describe and interpret experiences of moral challenges in everyday nursing practice, as expressed in reflection groups, and secondly, to further interpret how healthcare professionals’ self-understanding can bring light to such challenges. Data consisted of eight reflection group conversations with healthcare professionals, and were analysed using a phenomenological hermeneutic approach. Three themes have been formulated: ‘Having to be affected in order to help the patient’, ‘Having to accept that colleagues do not always collaborate’, and ‘Having to endure organizational demands’. The comprehensive understanding highlights that despite healthcare professionals’ struggle with demanding aspects of caring for their patients, they strive to do good. The study concludes that supporting healthcare professionals when they experience moral challenges is important particularly because such challenges seem to involve a challenge to healthcare professionals’ self-understanding, which may ultimately lead to their questioning why they are struggling with demanding situations in caring for the patient.

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.

Automorphism Groups of the Imprimitive Complex Reflection Groups II

2011 ◽  
Vol 18 (02) ◽  
pp. 315-326
Author(s):  
Li Wang
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Automorphism Groups ◽  
Reflection Group ◽  
Reflection Groups ◽  
Group Of Automorphisms ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Scalar Multiple

We prove that the automorphism group Aut (m,p,n) of an imprimitive complex reflection group G(m,p,n) is the product of a normal subgroup T(m,p,n) by a subgroup R(m,p,n), where R(m,p,n) is the group of automorphisms that preserve reflections and T(m,p,n) consists of automorphisms that map every element of G(m,p,n) to a scalar multiple of itself.

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