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

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.

Deformed diagonal harmonic polynomials for complex reflection groups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2897 ◽

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 ◽

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.

Gröbner-Shirshov Bases for Temperley-Lieb Algebras of Complex Reflection Groups

Symmetry ◽

10.3390/sym10100438 ◽

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 ◽

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

Algebra Colloquium ◽

10.1142/s1005386711000216 ◽

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 ◽

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.

Major Indices and Perfect Bases for Complex Reflection Groups

The Electronic Journal of Combinatorics ◽

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 ◽

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.

Quasi-invariants of complex reflection groups

Compositio Mathematica ◽

10.1112/s0010437x10005063 ◽

2010 ◽

Vol 147 (3) ◽

pp. 965-1002 ◽

Cited By ~ 5

Author(s):

Yuri Berest ◽

Oleg Chalykh

Keyword(s):

Differential Operators ◽

Reflection Group ◽

Reflection Groups ◽

Cherednik Algebras ◽

Complex Reflection ◽

Shift Operators ◽

Morita Equivalent ◽

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.

Models and refined models for involutory reflection groups and classical Weyl groups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2864 ◽

2010 ◽

Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽

Author(s):

Fabrizio Caselli ◽

Roberta Fulci

Keyword(s):

Weyl Group ◽

Coxeter Groups ◽

Finite Subgroup ◽

Weyl Groups ◽

Reflection Groups ◽

Complex Conjugation ◽

Complex Reflection ◽

Combinatorial Model ◽

International Audience

International audience A finite subgroup $G$ of $GL(n,\mathbb{C})$ is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group, i.e. elements $g \in G$ such that $g \bar{g}=1$, where the bar denotes complex conjugation. A uniform combinatorial model is constructed for all non-exceptional irreducible complex reflection groups which are involutory including, in particular, all infinite families of finite irreducible Coxeter groups. If $G$ is a classical Weyl group this result is much refined in a way which is compatible with the Robinson-Schensted correspondence on involutions. Un sous-groupe fini $G$ de GL(n,ℂ) est dit involutoire si la somme des dimensions de ses représentations irréductibles complexes est donné par le nombre de involutions absolues dans le groupe, c'est-a-dire le nombre de éléments $g \in G$ tels que $g \bar{g}=1$, où le bar dénote la conjugaison complexe. Un modèle combinatoire uniforme est construit pour tous les groupes de réflexions complexes irréductibles qui sont involutoires, en comprenant, toutes les familles de groupes de Coxeter finis irréductibles. Si $G$ est un groupe de Weyl ce résultat peut se raffiner d'une manière compatible avec la correspondance de Robinson-Schensted sur les involutions.

Submaximal factorizations of a Coxeter element in complex reflection groups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2955 ◽

2011 ◽

Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽

Author(s):

Vivien Ripoll

Keyword(s):

Reflection Group ◽

Coxeter Element ◽

Partition Lattice ◽

Combinatorial Object ◽

Noncrossing Partition ◽

Finite Reflection Group ◽

International Audience ◽

Cas A ◽

Noncrossing Partition Lattice

International audience When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $NC(W)$ as a generalized Fuß-Catalan number, depending on the invariant degrees of $W$. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $NC(W)$ as fibers of a "Lyashko-Looijenga covering''. This covering is constructed from the geometry of the discriminant hypersurface of $W$. We deduce new enumeration formulas for certain factorizations of a Coxeter element of $W$. Lorsque $W$ est un groupe de réflexion fini, le treillis $NC(W)$ des partitions non-croisées de type $W$ est un objet combinatoire très riche, qui généralise la notion de partitions non-croisées d'un $n$-gone. Une formule (seulement prouvée au cas par cas à l'heure actuelle) exprime le nombre de chaînes de longueur donnée dans $NC(W)$ sous la forme d'un nombre de Fuß-Catalan généralisé, qui dépend des degrés invariants de $W$. Nous décrivons une stratégie visant à comprendre certaines spécifications de cette formule de manière uniforme, en utilisant une interprétation des chaînes de $NC(W)$ comme fibres d'un "revêtement de Lyashko-Looijenga''. Ce revêtement est construit à partir de la géométrie de l'hypersurface du discriminant de $W$. Nous en déduisons de nouvelles formules de comptage pour certaines factorisations d'un élément de Coxeter de $W$.

Factorization problems in complex reflection groups

Canadian Journal of Mathematics ◽

10.4153/s0008414x2000022x ◽

2020 ◽

pp. 1-48

Author(s):

Joel Brewster Lewis ◽

Alejandro H. Morales

Keyword(s):

Space Dimension ◽

Reflection Group ◽

Coxeter Element ◽

Reflection Groups ◽

Complex Reflection ◽

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.

On the -problem for restrictions of complex reflection arrangements

Compositio Mathematica ◽

10.1112/s0010437x19007796 ◽

2020 ◽

Vol 156 (3) ◽

pp. 526-532

Author(s):

Nils Amend ◽

Pierre Deligne ◽

Gerhard Röhrle

Keyword(s):

Reflection Group ◽

Reflection Groups ◽

Complex Reflection ◽

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.

The Absolute Orders on the Coxeter Groups $A_n$ and $B_n$ are Sperner

The Electronic Journal of Combinatorics ◽

10.37236/8874 ◽

2020 ◽

Vol 27 (3) ◽

Author(s):

Lawrence H. Harper ◽

Gene B. Kim ◽

Neal Livesay

Keyword(s):

Coxeter Groups ◽

Symmetric Groups ◽

Product Theorem ◽

Reflection Groups ◽

Dihedral Groups ◽

Alternate Proof ◽

Sperner Property ◽

Noncrossing Partition ◽

The Absolute ◽

Partition Lattices

There are several classes of ranked posets related to reflection groups which are known to have the Sperner property, including the Bruhat orders and the generalized noncrossing partition lattices (i.e., the maximal intervals in absolute orders). In 2019, Harper–Kim proved that the absolute orders on the symmetric groups are (strongly) Sperner. In this paper, we give an alternate proof that extends to the signed symmetric groups and the dihedral groups. Our simple proof uses techniques inspired by Ford–Fulkerson's theory of networks and flows, and a product theorem.