scholarly journals Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r, p, n)

2010 ◽  
Vol 53 (2) ◽  
pp. 419-445 ◽  
Author(s):  
Stephen Griffeth
Keyword(s):  
Elementary Proof ◽  
Algebraic Combinatorics ◽  
Intertwining Operators ◽  
Reflection Groups ◽  
Orthogonal Functions ◽  
Cherednik Algebra ◽  
Complex Reflection Groups ◽  
Rational Cherednik Algebra ◽  
Invariant Ring ◽  
Combinatorial Representation Theory

AbstractThis paper aims to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category $\mathcal{O}$ for the rational Cherednik algebra of type G(r, p, n). As a first application, a self-contained and elementary proof of the analogue for the groups G(r, p, n), with r > 1, of Gordon's Theorem (previously Haiman's Conjecture) on the diagonal co-invariant ring is given. No restriction is imposed on p; the result for p ≠ r has been proved by Vale using a technique analogous to Gordon's. Because of the combinatorial application to Haiman's Conjecture, the paper is logically self-contained except for standard facts about complex reflection groups. The main results should be accessible to mathematicians working in algebraic combinatorics who are unfamiliar with the impressive range of ideas used in Gordon's proof of his theorem.

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 ON THE SMOOTHNESS OF CENTRES OF RATIONAL CHEREDNIK ALGEBRAS IN POSITIVE CHARACTERISTIC

2013 ◽  
Vol 55 (A) ◽  
pp. 27-54 ◽  
Author(s):  
GWYN BELLAMY ◽  
MAURIZIO MARTINO
Keyword(s):  
Deformation Parameter ◽  
Positive Characteristic ◽  
Reflection Group ◽  
Block Decomposition ◽  
Reflection Groups ◽  
Cherednik Algebras ◽  
Cherednik Algebra ◽  
Finite Dimensional ◽  
Rational Cherednik Algebra ◽  
Rational Cherednik Algebras

AbstractIn this paper we study rational Cherednik algebras att= 1 in positive characteristic. We study a finite-dimensional quotient of the rational Cherednik algebra called the restricted rational Cherednik algebra. When the corresponding pseudo-reflection group belongs to the infinite seriesG(m, d, n), we describe explicitly the block decomposition of the restricted algebra. We also classify all pseudo-reflection groups for which the centre of the corresponding rational Cherednik algebra is regular for generic values of the deformation parameter.

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

The Hilbert series of the irreducible quotient of the polynomial representation of the rational Cherednik algebra of type An−1 in characteristic p for p|n − 1

2021 ◽  
pp. 2250191
Author(s):  
Merrick Cai ◽  
Daniil Kalinov
Keyword(s):  
Hilbert Series ◽  
Positive Characteristic ◽  
Closed Field ◽  
Simple Criterion ◽  
Characteristic P ◽  
Polynomial Representation ◽  
Cherednik Algebra ◽  
Rational Cherednik Algebra ◽  
Polynomial Formula ◽  
Field Of Positive Characteristic

In this paper, we study the irreducible quotient [Formula: see text] of the polynomial representation of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] over an algebraically closed field of positive characteristic [Formula: see text] where [Formula: see text]. In the [Formula: see text] case, for all [Formula: see text] we give a complete description of the polynomials in the maximal proper graded submodule [Formula: see text], the kernel of the contravariant form [Formula: see text], and subsequently find the Hilbert series of the irreducible quotient [Formula: see text]. In the [Formula: see text] case, we give a complete description of the polynomials in [Formula: see text] when the characteristic [Formula: see text] and [Formula: see text] is transcendental over [Formula: see text], and compute the Hilbert series of the irreducible quotient [Formula: see text]. In doing so, we prove a conjecture due to Etingof and Rains completely for [Formula: see text], and also for any [Formula: see text] and [Formula: see text]. Furthermore, for [Formula: see text], we prove a simple criterion to determine whether a given polynomial [Formula: see text] lies in [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text] fixed.

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.

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