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 Champ: a Cherednik algebraMagmapackage

2015 ◽  
Vol 18 (1) ◽  
pp. 266-307 ◽  
Author(s):  
U. Thiel
Keyword(s):  
Las Vegas ◽  
Module Structure ◽  
Verma Modules ◽  
Reflection Groups ◽  
Cherednik Algebras ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Rational Cherednik Algebra ◽  
Las Vegas Algorithm ◽  
Rational Cherednik Algebras

We present a computer algebra package based onMagmafor performing computations in rational Cherednik algebras with arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero, which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We can determine the decomposition matrices of the Verma modules, the graded$G$-module structure of the simple modules, and the Calogero–Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we can also confirm Martino’s conjecture for several exceptional complex reflection groups.Supplementary materials are available with this article.

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 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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