Markov traces on affine and cyclotomic Yokonuma-Hecke algebras

In this article, we define and study the affine and cyclotomic Yokonuma-Hecke algebras. These algebras generalise at the same time the Ariki-Koike and affine Hecke algebras and the Yokonuma-Hecke algebras. We study the representation theory of these algebras and construct several bases for them. We then show how we can define Markov traces on them, which we in turn use to construct invariants for framed and classical knots in the solid torus. Finally, we study the Markov trace with zero parameters on the cyclotomic Yokonuma-Hecke algebras and determine the Schur elements with respect to that trace.Comment: 37 page

Tours de groupes et diagrammes de Bratteli

A new approach of the representation theory of the symmetric group has been developped by Okounkov et Vershik; it provides a different point of view on the subject compared to the "traditional" approaches. Moreover it intends to be generalizable to other chains of groups and algebras such as the other series of finite Coxeter groups, or the local and stationary towers of algebras. In this report we remind the usual presentation of the symmetric group and outline the new approach of its representation theory. Then we recall the definition of a local and stationary tower of algebras, of Bratteli diagram and of what is called Gelfand-Tsetlin basis and Gelfand-Tsetlin algebra in the new approach. We attempted to generalize this new approach to the chain of the alternating groups. For this purpose we study two known presentations of the alternating group, and give a new one which supplies the chain of the alternating groups with a structure of a local and stationary tower; in each case we realize the Coxeter-Todd algorithm and give a normal form for the elements of the group. We also begin the search for analogues of Jucys-Murphy elements for the alternating groups.Une nouvelle approche pour la théorie des représentations du groupe symétrique a été développée par Okounkov et Vershik ; elle fournit un éclairage différent sur le sujet par rapport aux approches « traditionnelles ». Par ailleurs, cette méthode vise à établir un cadre reproductible pour étudier les représentations d'autres chaînes de groupes et d'algèbres, telles que les autres séries de groupes de Coxeter finis, ou les tours d'algèbres locales et stationnaires. Dans ce mémoire, nous rappelons la présentation traditionnelle du groupe symétrique, et résumons le fond de la nouvelle approche pour sa théorie des représentations. Ensuite, nous rappelons la définition de tour d'algèbres locale et stationnaire, de diagramme de Bratteli et de ce qui est appelée base de Gelfand-Tsetlin et algèbre de Gelfand-Tsetlin dans la nouvelle approche. Nous nous sommes intéressés à la possibilité de généraliser cette approche pour la chaîne des groupes alternés. Dans ce but, nous étudions les présentations usuelles du groupe alterné, et en donnons une nouvelle qui munit la chaîne des groupes alternés d'une structure de tour locale et stationnaire ; dans chaque cas, nous réalisons l'algorithme de Coxeter-Todd et donnons une forme normale pour les éléments du groupe. Nous entamons également la recherche d'analogues des éléments de Jucys-Murphy pour les groupes alternés

Representation theory of the Yokonuma-Hecke algebra

We develop an inductive approach to the representation theory of the Yokonuma-Hecke algebra ${\rm Y}_{d,n}(q)$, based on the study of the spectrum of its Jucys-Murphy elements which are defined here. We give explicit formulas for the irreducible representations of ${\rm Y}_{d,n}(q)$ in terms of standard $d$-tableaux; we then use them to obtain a semisimplicity criterion. Finally, we prove the existence of a canonical symmetrising form on ${\rm Y}_{d,n}(q)$ and calculate the Schur elements with respect to that form.Comment: 28 page

Fusion Procedure for Cyclotomic Hecke Algebras

A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element

Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2)

In the spirit of the Schur-Weyl duality, we study the connections between the Racah algebra and the centralizers of tensor products of three (possibly different) irreducible representations of su(2). As a first step we show that the Racah algebra always surjects onto the centralizer. We then offer a conjecture regarding the description of the kernel of the map, which depends on the irreducible representations. If true, this conjecture would provide a presentation of the centralizer as a quotient of the Racah algebra. We prove this conjecture in several cases. In particular, while doing so, we explicitly obtain the Temperley-Lieb algebra, the Brauer algebra and the one-boundary Temperley-Lieb algebra as quotients of the Racah algebra

Morita equivalences for cyclotomic Hecke algebras of type B and D

42 pagesWe give a Morita equivalence theorem for so-called cyclotomic quotients of affine Hecke algebras of type B and D, in the spirit of a classical result of Dipper-Mathas in type A for Ariki-Koike algebras. As a consequence, the representation theory of affine Hecke algebras of type B and D reduces to the study of their cyclotomic quotients with eigenvalues in a single orbit under multiplication by $q^2$ and inversion. The main step in the proof consists in a decomposition theorem for generalisations of quiver Hecke algebras that appeared recently in the study of affine Hecke algebras of type B and D. This theorem reduces the general situation of a disconnected quiver with involution to a simpler setting. To be able to treat types B and D at the same time we unify the different definitions of generalisations of quiver Hecke algebra for type B that exist in the literature

Matrix elements of SO(3)SO(3) in sl3sl_3 representations as bispectral multivariate functions

We consider the change of basis under an $SO(3)$ automorphism for a finite-dimensional irreducible representation of $sl_3$. The coefficients of the transition matrix are expressed in terms of a double sum of products of Krawtchouk and Racah polynomials. These functions generalize the Griffiths-Krawtchouk polynomials, which are essentially the overlap coefficients in the case of symmetric representations of $sl_3$. Their recurrence and difference relations are obtained as byproducts of our construction. The proof is based on the decomposition of a general three-dimensional rotation in terms of elementary planar rotations and a transition between two embeddings of $sl_2$ in $sl_3$. The former is related to monovariate Krawtchouk polynomials and the latter, to monovariate Racah polynomials. The appearance of Racah polynomials in this context is algebraically explained by showing that the two $sl_2$ Casimir elements related to the two embeddings of $sl_2$ in $sl_3$ obey the Racah algebra relations. We also show that these two elements generate the centralizer in $U(sl_3)$ of the Cartan subalgebra and its complete algebraic description is given.Comment: 20 page

The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms

We propose a definition by generators and relations of the rank $n-2$ Askey-Wilson algebra $\mathfrak{aw}(n)$ for any integer $n$, generalising the known presentation for the usual case $n=3$. The generators are indexed by connected subsets of $\{1,\dots,n\}$ and the simple and rather small set of defining relations is directly inspired from the known case of $n=3$. Our first main result is to prove the existence of automorphisms of $\mathfrak{aw}(n)$ satisfying the relations of the braid group on $n+1$ strands. We also show the existence of coproduct maps relating the algebras for different values of $n$. An immediate consequence of our approach is that the Askey-Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the $n$-fold tensor product of the quantum group ${\rm U}_q(\mathfrak{sl}_2)$ or, equivalently, onto the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. We also obtain a family of central elements of the Askey-Wilson algebras which are shown, as a direct by-product of our construction, to be sent to $0$ in the realisation in the $n$-fold tensor product of ${\rm U}_q(\mathfrak{sl}_2)$, thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements

Algèbres de Hecke cyclotomiques : représentations, fusion et limite classique

An inductive approach to the representation theory of the chain of the cyclotomic Hecke algebras of type G(m,1,n) is developed. This approach relies on the study of the spectrum of a maximal commutative family formed by the analogues of the Jucys-Murphy elements. The irreducible representations, labelled by the multi-partitions, are constructed with the help of a new associative algebra, whose underlying vector space is the tensor product of the cyclotomic Hecke algebra with the free associative algebra generated by the standard multi-tableaux. The analogue of this approach is presented for the classical limit, that is for the chain of complex reflection groups of type G(m,1,n). In a second part, a basis of the cyclotomic Hecke algebras is given and the flatness of the deformation is proved without using the representation theory. These results are extended to the affine Hecke algebras of type A. Then a fusion procedure is presented for the complex reflection groups and the cyclotomic Hecke algebras of type G(m,1,n). In both cases, a complete set of primitive orthogonal idempotents is obtained by successive evaluations of a rational fonction. In a third part, a new presentation is obtained for the alternating subgroups of all Coxeter groups. The generators are related to oriented edges of the Coxeter graph. This presentation is then extended, for all types, to the spinor extensions of the alternating groups, the alternating Hecke algebras and the alternating subgroups of braid groups.Une approche inductive est développée pour la théorie des représentations de la chaîne des algèbres de Hecke cyclotomiques de type G(m,1,n). Cette approche repose sur l'étude du spectre d'une famille commutative maximale, formée par les analogues des éléments de Jucys-Murphy. Les représentations irréductibles, paramétrées par les multi-partitions, sont construites avec l'aide d'une nouvelle algèbre associative, dont l'espace vectoriel sous-jacent est le produit tensoriel de l'algèbre de Hecke cyclotomique avec l'algèbre associative libre engendrée par les multi-tableaux standards. L'analogue de cette approche est présentée pour la limite classique, c'est-à-dire la chaîne des groupes de réflexions complexes de type G(m,1,n). Dans une seconde partie, une base des algèbres de Hecke cyclotomiques est donnée et la platitude de la déformation est montrée sans utiliser la théorie des représentations. Ces résultats sont généralisés aux algèbres de Hecke affines de type A. Ensuite, une procédure de fusion est présentée pour les groupes de réflexions complexes et les algèbres de Hecke cyclotomiques de type G(m,1,n). Dans les deux cas, un ensemble complet d'idempotents primitifs orthogonaux est obtenu par évaluation consécutive d'une fonction rationnelle. Dans une troisième partie, une nouvelle présentation est obtenue pour les sous-groupes alternés de tous les groupes de Coxeter. Les générateurs sont reliés aux arêtes orientées du graphe de Coxeter. Cette présentation est ensuite étendue, pour tous les types, aux extensions spinorielles des groupes alternés, aux algèbres de Hecke alternées et aux sous-groupes alternés des groupes de tresses