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 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 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 A Refined Count of Coxeter Element Reflection Factorizations

10.37236/7362 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Elise DelMas ◽  
Thomas Hameister ◽  
Victor Reiner
Keyword(s):  
Generating Function ◽  
Coxeter Element ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Coxeter Number ◽  
Simple Product

For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the numberof reflections used from each orbit of hyperplanes. The proof is case-by-case via the classification of well-generated groups. It implies a new expression for the Coxeter number, expressed via data coming from a hyperplane orbit; a case-free proof of this due to J. Michel is included.

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 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 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 GL(n, q)-analogues of factorization problems in the symmetric group

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Joel Brewster Lewis ◽  
Alejandro H. Morales
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Space Dimension ◽  
Elliptic Element ◽  
Long Cycle ◽  
Fixed Space Dimension ◽  
International Audience ◽  
Fixed Space ◽  
Number Of Cycles

International audience We consider GLn (Fq)-analogues of certain factorization problems in the symmetric group Sn: ratherthan counting factorizations of the long cycle(1,2, . . . , n) given the number of cycles of each factor, we countfactorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in Sn, the generating function counting these factorizations has attractive coefficients after an appropriate change of basis.Our work generalizes several recent results on factorizations in GLn (Fq) and also uses a character-based approach.We end with an asymptotic application and some questions.

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.

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