We develop an inductive approach to the representation theory of the Yokonuma-Hecke algebra 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 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 Y d,n (q) and calculate the Schur elements with respect to that form.2010 Mathematics Subject Classification. 20C08, 05E10, 16S80.The Yokonuma-Hecke algebra is quite different from the other famous deformation of the group algebra of G(d, 1, n), the Ariki-Koike algebra [ArKo]. As mentioned above, the Yokonuma-Hecke algebra is a quotient of the group algebra of the modular framed braid group (Z/dZ) ≀ B n . The Ariki-Koike algebra is a quotient of the group algebra of the braid group of type B. Thus, in some sense, the Yokonuma-Hecke algebra is a deformation where the wreath product structure of G(d, 1, n) is respected; in the case of the Ariki-Koike algebra this structure is lost in view of the preservation of the classical Hecke quadratic relation. On the other hand, this preservation of the quadratic relation makes the Iwahori-Hecke algebra of type A an obvious subalgebra of the Ariki-Koike algebra. As far as the Yokonuma-Hecke algebra is concerned, the Iwahori-Hecke algebra H n (q) of type A is an obvious quotient of Y d,n (q), but not an obvious subalgebra.In this paper, we will study the representation theory of this object of high algebraic and topological interest. Some information on its representation theory in the general context of unipotent Hecke algebras has been obtained by Thiem in [Th1, Th2, Th3]. Unfortunately, the generality of his results and the change of presentation for Y d,n (q) do not allow us to put this information to practical use. In this paper, we develop an inductive, and highly combinatorial, approach to the representation theory of the Yokonuma-Hecke algebra. We give explicit formulas for all its irreducible presentations (in the semisimple case), which one can work with. Thanks to our formulas we are able to obtain a semisimplicity criterion for Y d,n (q), and calculate its Schur elements with respect to the canonical symmetrising form defined here, thus getting a glimpse of the modular representation theory of the Yokonuma-Hecke algebra.For d = 1, the approach presented here coincides with the one in [IsOg] for the Iwahori-Hecke algebra of type A. For d ≥ 1, an inductive approach, in the spirit of [OkVe], was given in [OgPo2] for the complex reflection group G (d, 1, n). Our approach for the Yokonuma-Hecke algebra can be seen as a deformed version of the approach in [OgPo2] (different from the deformed version presented in [OgPo1] for the Ariki-Koike algebra).The paper is organised as follows: in Section 2, we define the Yokonuma-Hecke algebra Y d,n (q), and we suggest analogues for Y d,n (q) of the Jucys-Murphy elements of the symmetric group; these Jucys-Murphy ele...

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. We are grateful to Sofia Lambropoulou and Jesús Juyumaya for their useful suggestions and corrections to this paper. We would also like to thank Vincent Sécherre for our fruitful conversations and his interesting remarks concerning the affine case.

We develop several applications of the fact that the Yokonuma-Hecke algebra of the general linear group GL is isomorphic to a direct sum of matrix algebras associated to Iwahori-Hecke algebras of type A. This includes a description of the semisimple and modular representation theory of the Yokonuma-Hecke algebras of GL and a complete classification of all the Markov traces for them. Finally, from these Markov traces, we construct 3-variables polynomials which are invariants for framed and classical knots and links, and investigate their properties with the help of the isomorphism.