This paper is a presentation, where we compute the HOMFLYPT Skein module of singular links in the 3-sphere. This calculation is based on some results previously proved by Rabenda and the author on Markov traces on singular Hecke algebras, as well as on classical techniques that allow to pass from the framework of Markov traces on Hecke algebras to the framework of HOMFLYPT Skein modules. Some o...

Given a Hecke symmetry R, one can define a matrix bialgebra ER and a matrix Hopf algebra HR, which are called function rings on the matrix quantum semi-group and matrix quantum groups associated to R. We show that for an even Hecke symmetry, the rational representations of the corresponding quantum group are absolutely reducible and that the fusion coefficients of simple representations depend ...

We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q = −1. This allows us to define graded modules over the Hecke algebra at q = −1 that are ‘odd’ analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient...

In the previous part of this paper, we constructed a large family of Hecke algebras on some classical groups G de¢ned over p-adic ¢elds in order to understand their admissible representations. EachHecke algebra is associated to a pair …JS; rS†of an open compact subgroup JS and its irreducible representation rSwhich is constructed fromgiven dataS ˆ …G;P0 0; R†. Here, G is a semisimple element in...

We introduce a new class (in two versions, Au and Bu) of rational double affine Hecke algebras (DaHa) associated to the spin symmetric group. We establish the basic properties of the algebras, such as PBW and Dunkl representation, and connections to Nazarov’s degenerate affine Hecke algebra and to a new degenerate affine Hecke algebra introduced here. We formulate a precise connection between t...

Dedekind symbols generalize the classical Dedekind sums (symbols). The symbols are determined uniquely by their reciprocity laws up to an additive constant. There is a natural isomorphism between the space of Dedekind symbols with polynomial (Laurent polynomial) reciprocity laws and the space of cusp (modular) forms. In this article we introduce Hecke operators on the space of weighted Dedekind...

Abstract. Quantum Drinfeld Hecke algebras are generalizations of Drinfeld Hecke algebras in which polynomial rings are replaced by quantum polynomial rings. We identify these algebras as deformations of skew group algebras, giving an explicit connection to Hochschild cohomology. We compute the relevant part of Hochschild cohomology for actions of many reflection groups and we exploit computatio...

With any even Hecke symmetry R (that is a Hecke type solution of the Yang-Baxter equation) we associate a quasitensor category. We formulate a condition on R implying that the constructed category is rigid and its commutativity isomorphisms RU,V are natural in the sense of [T]. We show that this condition leads to rescaling the initial Hecke symmetry. We suggest a new way of introducing traces ...

Abstract. We investigate the combinatorial properties of the traces of the n-th Hecke operators on the spaces of weight 2k cusp forms of level N . We establish examples in which these traces are expressed in terms of classical objects in enumerative combinatorics (e.g. tilings and Motzkin paths). We establish in general that Hecke traces are explicit rational linear combinations of values of Ge...

We describe a recursive algorithm that produces an integral basis for the centre of the Hecke algebra of type A consisting of linear combinations of monomial symmetric polynomials of Jucys–Murphy elements. We also discuss the existence of integral bases for the centre of the Hecke algebra that consist solely of monomial symmetric polynomials of Jucys–Murphy elements. Finally, for n = 3, we show...