scholarly journals Dunkl operators and quasi-invariants of complex reflection groups

2012 ◽  
pp. 1-23 ◽  
Author(s):  
Yuri Berest ◽  
Peter Samuelson
Keyword(s):  
Reflection Groups ◽  
Dunkl Operators ◽  
Complex Reflection ◽  
Complex Reflection Groups

scholarly journals DUNKL OPERATORS FOR COMPLEX REFLECTION GROUPS

2003 ◽  
Vol 86 (1) ◽  
pp. 70-108 ◽  
Author(s):  
C. F. DUNKL ◽  
E. M. OPDAM
Keyword(s):  
Reflection Groups ◽  
Dunkl Operators ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Commuting Operators ◽  
Rham Complex ◽  
Rational Cherednik Algebra ◽  
De Rham ◽  
Parameterized Family ◽  
Primary 52C35

Dunkl operators for complex reflection groups are defined in this paper. These commuting operators give rise to a parameterized family of deformations of the polynomial De Rham complex. This leads to the study of the polynomial ring as a module over the ‘rational Cherednik algebra’, and a natural contravariant form on this module. In the case of the imprimitive complex reflection groups $G(m, p, N)$, the set of singular parameters in the parameterized family of these structures is described explicitly, using the theory of non-symmetric Jack polynomials.2000 Mathematical Subject Classification: 20F55 (primary), 52C35, 05E05, 33C08 (secondary).

scholarly journals Rouquier blocks of the cyclotomic Hecke algebras of G(de, e, r)

2010 ◽  
Vol 197 ◽  
pp. 175-212
Author(s):  
Maria Chlouveraki
Keyword(s):  
Infinite Series ◽  
Hecke Algebras ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Cyclotomic Hecke Algebras

The Rouquier blocks of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the families of characters defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite seriesG(de, e, r), thus completing their calculation for all complex reflection groups.

Complex reflection groups

1990 ◽  
Vol 18 (12) ◽  
pp. 3999-4029 ◽  
Author(s):  
M.C. Hughes
Keyword(s):  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups

scholarly journals Euler Characteristic of the Truncated Order Complex of Generalized Noncrossing Partitions

10.37236/232 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
D. Armstrong ◽  
C. Krattenthaler
Keyword(s):  
Euler Characteristic ◽  
Order Complex ◽  
Reflection Groups ◽  
Noncrossing Partitions ◽  
Minimal Elements ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Phd Thesis

The purpose of this paper is to complete the study, begun in the first author's PhD thesis, of the topology of the poset of generalized noncrossing partitions associated to real reflection groups. In particular, we calculate the Euler characteristic of this poset with the maximal and minimal elements deleted. As we show, the result on the Euler characteristic extends to generalized noncrossing partitions associated to well-generated complex reflection groups.

scholarly journals On Fomin–Kirillov algebras for complex reflection groups

2016 ◽  
Vol 45 (8) ◽  
pp. 3653-3666
Author(s):  
Robert Laugwitz
Keyword(s):  
Reflection Groups ◽  
Complex Reflection ◽  
Complex 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 Generalized Green functions associated to complex reflection groups

2020 ◽  
Vol 558 ◽  
pp. 677-707
Author(s):  
Toshiaki Shoji
Keyword(s):  
Green Functions ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups

scholarly journals An inductive approach to representations of complex reflection groups G(m, 1, n)

2013 ◽  
Vol 174 (1) ◽  
pp. 95-108 ◽  
Author(s):  
O. V. Ogievetsky ◽  
L. Poulain d’Andecy
Keyword(s):  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Inductive Approach

scholarly journals Noncommutative Noether’s problem for complex reflection groups

2017 ◽  
Vol 145 (12) ◽  
pp. 5043-5052 ◽  
Author(s):  
Farkhod Eshmatov ◽  
Vyacheslav Futorny ◽  
Sergiy Ovsienko ◽  
Joao Fernando Schwarz
Keyword(s):  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Noether’S Problem
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