scholarly journals Completely positive maps for imprimitive complex reflection groups

2021 ◽  
Vol 13 (2) ◽  
pp. 452-459
Author(s):  
H. Randriamaro
Keyword(s):  
Hilbert Space ◽  
Coxeter Groups ◽  
Fock Space ◽  
Inner Product ◽  
Reflection Groups ◽  
Bounded Operators ◽  
Completely Positive ◽  
Creation And Annihilation Operators ◽  
Complex Reflection ◽  
Complex Reflection Groups

In 1994, M. Bożejko and R. Speicher proved the existence of completely positive quasimultiplicative maps from the group algebra of Coxeter groups to the set of bounded operators. They used some of them to define an inner product associated to creation and annihilation operators on a direct sum of Hilbert space tensor powers called full Fock space. Afterwards, A. Mathas and R. Orellana defined in 2008 a length function on imprimitive complex reflection groups that allowed them to introduce an analogue to the descent algebra of Coxeter groups. In this article, we use the length function defined by A. Mathas and R. Orellana to extend the result of M. Bożejko and R. Speicher to imprimitive complex reflection groups, in other words to prove the existence of completely positive quasimultiplicative maps from the group algebra of imprimitive complex reflection groups to the set of bounded operators. Some of those maps are then used to define a more general inner product associated to creation and annihilation operators on the full Fock space. Recall that in quantum mechanics, the state of a physical system is represented by a vector in a Hilbert space, and the creation and annihilation operators act on a Fock state by respectively adding and removing a particle in the ascribed quantum state.

scholarly journals EMBEDDINGS OF COMPLEX LINE SYSTEMS AND FINITE REFLECTION GROUPS

2008 ◽  
Vol 85 (2) ◽  
pp. 211-228 ◽  
Author(s):  
MURALEEDARAN KRISHNASAMY ◽  
D. E. TAYLOR
Keyword(s):  
Infinite Family ◽  
Inner Product ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Closed Sets ◽  
Complex Reflection Groups ◽  
Least Eigenvalue ◽  
Third Line ◽  
Hermitian Inner Product

AbstractA star is a planar set of three lines through a common point in which the angle between each pair is 60∘. A set of lines through a point in which the angle between each pair of lines is 60 or 90∘ is star-closed if for every pair of its lines at 60∘ the set contains the third line of the star. In 1976 Cameron, Goethals, Seidel and Shult showed that the indecomposable star-closed sets in Euclidean space are the root systems of types An, Dn, E6, E7 and E8. This result was a key part of their determination of all graphs with least eigenvalue −2. Subsequently, Cvetković, Rowlinson and Simić determined all star-closed extensions of these line systems. We generalize this result on extensions of line systems to complex n-space equipped with a hermitian inner product. There is one further infinite family, and two exceptional types arising from Burkhardt and Mitchell’s complex reflection groups in dimensions five and six. The proof is a geometric version of Mitchell’s classification of complex reflection groups in dimensions greater than four.

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

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 ◽  
Complex Reflection Groups ◽  
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.

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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