Yokonuma Hecke Algebras Research Articles

Symplectic Fourier–Deligne Transforms on G/U and the Algebra of Braids and Ties

Abstract We explicitly identify the algebra generated by symplectic Fourier–Deligne transforms (i.e., convolution with Kazhdan–Laumon sheaves) acting on the Grothendieck group of perverse sheaves on the basic affine space $G/U$, answering a question originally raised by A. Polishchuk. We show it is isomorphic to a distinguished subalgebra, studied by I. Marin, of the generalized algebra of braids and ties (defined in Type $A$ by F. Aicardi and J. Juyumaya and generalized to all types by Marin), providing a connection between geometric representation theory and an algebra defined in the context of knot theory. Our geometric interpretation of this algebra entails some algebraic consequences: we obtain a short and type-independent geometric proof of the braid relations for Juyumaya’s generators of the Yokonuma–Hecke algebra (previously proved case-by-case in types $A, D, E$ by Juyumaya and separately for types $B, C, F_{4}, G_{2}$ by Juyumaya and S. S. Kannan), a natural candidate for an analogue of a Kazhdan–Lusztig basis, and finally an explicit formula for the dimension of Marin’s algebra in Type $A_{n}$ (previously only known for $n \leq 4$).

Framization of a Temperley-Lieb algebra of type B

We extend the Framization of the Temperley-Lieb algebra to Coxeter systems of type B. We first define a natural extension of the classical Temperley-Lieb algebra to Coxeter systems of type B and prove that such an extension supports a unique linear Markov trace function. We then introduce the Framization of the Temperley-Lieb algebra of type B as a quotient of the Yokonuma-Hecke algebra of type B. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra of type B to pass to the quotient algebra. Using the main theorem, we construct invariants for framed links and classical links inside the solid torus.

Modular Representation Theory of Affine and Cyclotomic Yokonuma-Hecke Algebras

We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type A (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic p. Secondly, the modular branching rules for these algebras are obtained; moreover, the resulting modular branching graphs for cyclotomic Yokonuma-Hecke algebras are identified with crystal graphs of irreducible integrable representations of affine Lie algebras of type A.

Arithmetic and representation theory of wild character varieties

We count points over a finite field on wild character varieties of Riemann surfaces for singularities with regular semisimple leading term. The new feature in our counting formulas is the appearance of characters of Yokonuma–Hecke algebras. Our result leads to the conjecture that the mixed Hodge polynomials of these character varieties agree with the previously conjectured perverse Hodge polynomials of certain twisted parabolic Higgs moduli spaces, indicating the possibility of a P=W conjecture for a suitable wild Hitchin system.

The HOMFLY-PT polynomials of sublinks and the Yokonuma–Hecke algebras

We describe completely the link invariants constructed using Markov traces on the Yokonuma–Hecke algebras in terms of the linking matrix and the Hoste–Ocneanu–Millett–Freyd–Lickorish–Yetter–Przytycki–Traczyk (HOMFLY-PT) polynomials of sublinks.

Identifying the Invariants for Classical Knots and Links from the Yokonuma–Hecke Algebras

Abstract We announce the existence of a family of new 2-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma–Hecke algebra of type A. Yokonuma–Hecke algebras are generalizations of Iwahori–Hecke algebras, and this family contains the HOMFLYPT polynomial, the famous 2-variable invariant for classical links arising from the Iwahori–Hecke algebra of type A. We show that these invariants are topologically equivalent to the HOMFLYPT polynomial on knots, but not on links, by providing pairs of HOMFLYPT-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new 3-variable skein link invariant that is stronger than the HOMFLYPT polynomial. Finally, we present a closed formula for this invariant, by W. B. R. Lickorish, that uses HOMFLYPT polynomials of sublinks and linking numbers of a given oriented link.

Cell structures for the Yokonuma–Hecke algebra and the algebra of braids and ties

We construct a faithful tensor representation for the Yokonuma–Hecke algebra Yr,n, and use it to give a concrete isomorphism between Yr,n and Shoji's modified Ariki–Koike algebra. We give a cellular basis for Yr,n and show that the Jucys–Murphy elements for Yr,n are JM-elements in the abstract sense. Finally, we construct a cellular basis for the Aicardi–Juyumaya algebra of braids and ties.

Lattice extensions of Hecke algebras

We investigate the extensions of the Hecke algebras of finite (complex) reflection groups by lattices of reflection subgroups that we introduced, for some of them, in our previous work on the Yokonuma–Hecke algebras and their connections with Artin groups. When the Hecke algebra is attached to the symmetric group, and the lattice contains all reflection subgroups, then these algebras are the diagram algebras of braids and ties of Aicardi and Juyumaya. We prove a structure theorem for these algebras, generalizing a result of Espinoza and Ryom-Hansen from the case of the symmetric group to the general case. We prove that these algebras are symmetric algebras at least when W is a Coxeter group, and in general under the trace conjecture of Broué, Malle and Michel.

Fusion procedure for Yokonuma–Hecke algebras

By generalizing the approach of Poulain d’Andecy (2014), we introduce certain Baxterized elements for Yokonuma–Hecke algebras, and use them to prove that a complete set of pairwise orthogonal primitive idempotents of Yokonuma–Hecke algebras can be obtaine

Affine cellularity of affine Yokonuma–Hecke algebras

We establish an explicit algebra isomorphism between the affine Yokonuma–Hecke algebra Yˆr,n(q) and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras of type A. As an application of this result, we show that Yˆr,n(q) is affine cellular in the sense of Koenig and Xi, and further prove that it has finite global dimension when the parameter q is not a root of the Poincaré polynomial. As another application, we also recover the modular representation theory of Yˆr,n(q) previously obtained in [7].

Skein Invariants of Links and Their State Sum Models

We present the new skein invariants of classical links, H [ H ] , K [ K ] and D [ D ] , based on the invariants of links, H, K and D, denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. The invariants are obtained by abstracting the skein relation of the corresponding invariant and making a new skein algorithm comprising two computational levels: first producing unlinked knotted components, then evaluating the resulting knots. The invariants in this paper, were revealed through the skein theoretic definition of the invariants Θ d related to the Yokonuma–Hecke algebras and their 3-variable generalization Θ , which generalizes the Homflypt polynomial. H [ H ] is the regular isotopy counterpart of Θ . The invariants K [ K ] and D [ D ] are new generalizations of the Kauffman and the Dubrovnik polynomials. We sketch skein theoretic proofs of the well-definedness and topological properties of these invariants. The invariants of this paper are reformulated into summations of the generating invariants (H, K, D) on sublinks of the given link L, obtained by partitioning L into collections of sublinks. The first such reformulation was achieved by W.B.R. Lickorish for the invariant Θ and we generalize it to the Kauffman and Dubrovnik polynomial cases. State sum models are formulated for all the invariants. These state summation models are based on our skein template algorithm which formalizes the skein theoretic process as an analogue of a statistical mechanics partition function. Relationships with statistical mechanics models are articulated. Finally, we discuss physical situations where a multi-leveled course of action is taken naturally.

Clifford theory for Yokonuma-Hecke algebras and deformation of complex reflection groups

We define and study an action of the symmetric group on the Yokonuma--Hecke algebra. This leads to the definition of two classes of algebras. The first one is connected with the image of the algebra of the braid group inside the Yokonuma--Hecke algebras, and in turn with an algebra defined by Aicardi and Juyumaya known as the algebra of braids and ties. The second one can be seen as new deformations of complex reflection groups of type G(d,p,n). We provide several presentations for both algebras and a complete study of their representation theories using Clifford theory.

Classical link invariants from the framizations of the Iwahori–Hecke algebra and the Temperley–Lieb algebra of type A

In this paper, we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma–Hecke algebras [Formula: see text], which are not topologically equivalent to the Homflypt polynomial. We then present the algebra [Formula: see text] which is the appropriate Temperley–Lieb analogue of [Formula: see text], as well as the related 1-variable classical link invariants, which in turn are not topologically equivalent to the Jones polynomial. Finally, we present the algebra of braids and ties which is related to the Yokonuma–Hecke algebra, and also its quotient, the partition Temperley–Lieb algebra [Formula: see text] and we prove an isomorphism of this algebra with a subalgebra of [Formula: see text].

Cyclotomic Yokonuma–Hecke algebras are cyclotomic quiver Hecke algebras

We prove that cyclotomic Yokonuma–Hecke algebras of type A are cyclotomic quiver Hecke algebras and we give an explicit isomorphism with its inverse, using a similar result of Brundan and Kleshchev on cyclotomic Hecke algebras. The quiver we use is given by disjoint copies of cyclic quivers. We relate this work to an isomorphism of Lusztig.

Artin Groups and Yokonuma–Hecke Algebras

We attach to every Coxeter system (W,S) an extension C_W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C_W. When W is finite, we prove that this algebra is a free module of finite rank which is generically semisimple. When W is the Weyl group of a Chevalley group, C_W naturally maps to the associated Yokonuma-Hecke algebra. When W = S_n this algebra can be identified with a diagram algebra called the algebra of `braids and ties'. The image of the usual braid group in this case is investigated. Finally, we generalize our construction to finite complex reflection groups, thus extending the Broue-Malle-Rouquier construction of a generalized Hecke algebra attached to these groups.

Framization of the Temperley–Lieb algebra

We propose a framization of the Temperley-Lieb algebra. The framization is a procedure that can briefly be described as the adding of framing to a known knot algebra in a way that is both algebraically consistent and topologically meaningful. Our framization of the Temperley-Lieb algebra is defined as a quotient of the Yokonuma-Hecke algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra to pass through to the quotient algebra. Using this we construct 1-variable invariants for classical knots and links, which, as we show, are not topologically equivalent to the Jones polynomial.

Enabling computations for link invariants coming from the Yokonuma–Hecke algebras

We describe an algorithm for computing the invariants of classical links arising from the Yokonuma–Hecke algebras. A detailed overview of the algorithm is given, following closely its implementation, a program used to calculate the invariants on several Homflypt-equivalent pairs of links.

On the link invariants from the Yokonuma–Hecke algebras

In this paper, we study properties of the Markov trace tr[Formula: see text] and the specialized trace [Formula: see text] on the Yokonuma–Hecke algebras, such as behavior under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace [Formula: see text] and also invariants for transverse links through the trace tr[Formula: see text]. In order to compare the invariants for classical links with the Homflypt polynomial, we develop computer programs and we evaluate them on several Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behavior on links.

The affine Yokonuma–Hecke algebra and the pro-$p$-Iwahori–Hecke algebra

We prove that the affine Yokonuma-Hecke algebra defined by Chlouveraki and Poulain d'Andecy is a particular case of the pro-$p$-Iwahori-Hecke algebra defined by Vign\'eras.

An isomorphism theorem for Yokonuma–Hecke algebras and applications to link invariants

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. In particular, for classical knots, a consequence of the construction is that the obtained set of invariants is topologically equivalent to the HOMFLYPT polynomial. We thus recover results of Chlouveraki et al. (2015, arXiv:1505.06666) about the Juyumaya–Lambropoulou invariants.