scholarly journals Major Indices and Perfect Bases for Complex Reflection Groups

10.37236/785 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Robert Shwartz ◽  
Ron M. Adin ◽  
Yuval Roichman
Keyword(s):  
Direct Product ◽  
Hilbert Series ◽  
Reflection Group ◽  
Reflection Groups ◽  
Cyclic Subgroups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Mild Conditions ◽  
Major Index ◽  
Complex Reflection Group

It is shown that, under mild conditions, a complex reflection group $G(r,p,n)$ may be decomposed into a set-wise direct product of cyclic subgroups. This property is then used to extend the notion of major index and a corresponding Hilbert series identity to these and other closely related 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.

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.

scholarly journals Quasi-invariants of complex reflection groups

2010 ◽  
Vol 147 (3) ◽  
pp. 965-1002 ◽  
Author(s):  
Yuri Berest ◽  
Oleg Chalykh
Keyword(s):  
Differential Operators ◽  
Reflection Group ◽  
Reflection Groups ◽  
Cherednik Algebras ◽  
Complex Reflection ◽  
Shift Operators ◽  
Complex Reflection Groups ◽  
Morita Equivalent ◽  
Complex Reflection Group ◽  
Rings Of Differential Operators

AbstractWe introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasi-invariants of a given multiplicity is not, in general, an algebra but a module Qk over the coordinate ring of a (singular) affine variety Xk. We extend the main results of Berest et al. [Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279–337] to this setting: in particular, we show that the variety Xk and the module Qk are Cohen–Macaulay, and the rings of differential operators on Xk and Qk are simple rings, Morita equivalent to the Weyl algebra An(ℂ) , where n=dim Xk. Our approach relies on representation theory of complex Cherednik algebras introduced by Dunkl and Opdam [Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), 70–108] and is parallel to that of Berest et al. As an application, we prove the existence of shift operators for an arbitrary complex reflection group, confirming a conjecture of Dunkl and Opdam. Another result is a proof of a conjecture of Opdam, concerning certain operations (KZ twists) on the set of irreducible representations of W.

scholarly journals Factorization problems in complex reflection groups

2020 ◽  
pp. 1-48
Author(s):  
Joel Brewster Lewis ◽  
Alejandro H. Morales
Keyword(s):  
Space Dimension ◽  
Reflection Group ◽  
Coxeter Element ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Fixed Space Dimension ◽  
Fixed Space

Abstract We enumerate factorizations of a Coxeter element in a well-generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of W-noncrossing partitions.

scholarly journals On the -problem for restrictions of complex reflection arrangements

2020 ◽  
Vol 156 (3) ◽  
pp. 526-532
Author(s):  
Nils Amend ◽  
Pierre Deligne ◽  
Gerhard Röhrle
Keyword(s):  
Reflection Group ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Reflection Arrangement

Let $W\subset \operatorname{GL}(V)$ be a complex reflection group and $\mathscr{A}(W)$ the set of the mirrors of the complex reflections in $W$. It is known that the complement $X(\mathscr{A}(W))$ of the reflection arrangement $\mathscr{A}(W)$ is a $K(\unicode[STIX]{x1D70B},1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr{A}(W)$, let $X(\mathscr{A}(W)^{Y})$ be the complement in $Y$ of the hyperplanes in $\mathscr{A}(W)$ not containing $Y$. We hope that $X(\mathscr{A}(W)^{Y})$ is always a $K(\unicode[STIX]{x1D70B},1)$. We prove it in case of the monomial groups $W=G(r,p,\ell )$. Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(\unicode[STIX]{x1D70B},1)$ property remains to be proved.

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.

scholarly journals AUTOMORPHISM GROUPS OF THE IMPRIMITIVE COMPLEX REFLECTION GROUPS

2009 ◽  
Vol 86 (1) ◽  
pp. 123-138 ◽  
Author(s):  
JIAN-YI SHI ◽  
LI WANG
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Reflection Group ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group

AbstractWe describe the group of all reflection-preserving automorphisms of an imprimitive complex reflection group. We also study some properties of this automorphism group.

scholarly journals Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Henri Mühle
Keyword(s):  
Coxeter Groups ◽  
Reflection Group ◽  
Sperner Property ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Complex Reflection Group ◽  
Noncrossing Partition ◽  
International Audience ◽  
Partition Lattices ◽  
Special Case

International audience We prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B.

scholarly journals NORMAL REFLECTION SUBGROUPS OF COMPLEX REFLECTION GROUPS

2021 ◽  
pp. 1-39
Author(s):  
Carlos E. Arreche ◽  
Nathan F. Williams
Keyword(s):  
Generating Function ◽  
Space Dimension ◽  
Reflection Group ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Fixed Space Dimension ◽  
Normal Reflection ◽  
Recent Theorem ◽  
Fixed Space

Abstract We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalised exponents. Our refinement gives a uniform proof and generalisation of a recent theorem of the second author.

scholarly journals Enumerating projective reflection groups

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Riccardo Biagioli ◽  
Fabrizio Caselli
Keyword(s):  
Representation Theory ◽  
Generating Functions ◽  
Special Class ◽  
Hilbert Series ◽  
Weyl Groups ◽  
Reflection Groups ◽  
One Dimensional ◽  
Complex Reflection Groups ◽  
Major Index ◽  
International Audience

International audience Projective reflection groups have been recently defined by the second author. They include a special class of groups denoted G(r,p,s,n) which contains all classical Weyl groups and more generally all the complex reflection groups of type G(r,p,n). In this paper we define some statistics analogous to descent number and major index over the projective reflection groups G(r,p,s,n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r,p,s,n), as distribution of one-dimensional characters and computation of Hilbert series of some invariant algebras, are also treated. Les groupes de réflexions projectifs ont été récemment définis par le deuxième auteur. Ils comprennent une classe spéciale de groupes notée G(r,p,s,n), qui contient tous les groupes de Weyl classiques et plus généralement tous les groupes de réflexions complexes du type G(r,p,n). Dans ce papier on définit des statistiques analogues au nombre de descentes et à l'indice majeur pour les groupes G(r,p,s,n), et on calcule plusieurs fonctions génératrices. Certains aspects de la théorie des représentations de G(r,p,s,n), comme la distribution des caractères linéaires et le calcul de la série de Hilbert de quelques algèbres d'invariants, sont aussi abordés.

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