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Given a Hecke symmetry R, one can define a matrix bialgebra E_R and a matrix Hopf algebra H_R, 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 only on the rank of the Hecke symmetry. Further we compute the quantum rank of simple representations. We also show that the quantum semi-group is "Zariski" dense in the quantum group. Finally we give a formula for the integral.

Kazhdan and Wenzl classified all rigid tensor categories with fusion ring isomorphic to the fusion ring of the group SU(d). In this paper we consider the C*-analogue of this problem. Given a rigid C*-tensor category 𝒞 with fusion ring isomorphic to the fusion ring of the group SU(d), we can extract a constant q from 𝒞 such that there exists a *-representation of the Hecke algebra H_n(q) into 𝒞. The categorical trace on 𝒞 induces a Markov trace on H_n(q). Using this Markov trace and a representation of H_n(q) in we show that 𝒞 is equivalent to a twist of the category . Furthermore a sufficient condition on a C*-tensor category 𝒞 is given for existence of an embedding of a twist of in 𝒞.

Let $U{(n)}_{{𝒵}}$ be the Lusztig integral form of quantum ${𝔤𝔩}_n$. There is a natural surjective algebra homomorphism ${\zeta }_{r,{𝒵}}$ from $U{(n)}_{{𝒵}}$ to the integral $q$-Schur algebra ${𝒮}{(n,r)}_{{𝒵}}$. We give a generating set for the kernel of ${\zeta }_{r,{𝒵}}$. In particular, we obtain a presentation of the $q$-Schur algebra by generators and relations over any field.

An approach, based on Jucys–Murphy elements, to the representation theory of cyclotomic Hecke algebras is developed. The maximality (in the cyclotomic Hecke algebra) of the set of the Jucys–Murphy elements is established. A basis of the cyclotomic Hecke algebra is suggested; this basis is used to establish the flatness of the deformation without using the representation theory.

This article is devoted to the study of several algebras related to symmetric functions, which admit linear bases labelled by various combinatorial objects: permutations (free quasi-symmetric functions), standard Young tableaux (free symmetric functions) and packed integer matrices (matrix quasi-symmetric functions). Free quasi-symmetric functions provide a kind of noncommutative Frobenius characteristic for a certain category of modules over the 0-Hecke algebras. New examples of indecomposable H_n(0)-modules are discussed, and the homological properties of H_n(0) are computed for small n. Finally, the algebra of matrix quasi-symmetric functions is interpreted as a convolution algebra.

The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements whose sum generates the centre. They can be represented by simple tangles in the Homfly skein theory version of H_n. In this paper I present a single tangle which represents their sum, and which is obviously central. As a consequence it is possible to identify a natural basis for the Homfly skein of the annulus, .

Symmetric functions of the Murphy operators are also central in H_n. I define geometrically a homomorphism from to the centre of each algebra H_n, and find an element in , independent of n, whose image is the mth power sum of the Murphy operators. Generating function techniques are used to describe images of other elements of in terms of the Murphy operators, and to demonstrate relations among other natural skein elements.

The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Q_λ, depending on partitions λ. We show how the 2-variable Homfly invariant <λ, μ> of the Hopf link arising from decorations Q_λ and Q_μ can be found from the Schur symmetric function s_μ of an explicit power series depending on λ. We show also that the quantum invariant of the Hopf link coloured by irreducible sl(N)_q modules V_λ and V_μ, which is a 1-variable specialisation of <λ, μ>, can be expressed in terms of an N × N minor of the Vandermonde matrix (q^ij).

We use a skien-theoretic version of the Hecke algebras of type A to present three-dimensional diagrammatic views of Gyoja's idempotent elements, based closely on the corresponding Young diagram λ. In this context we give straightforward calculations for the eigenvalues f_λ and m_λ of two natural central elements in the Hecke algebras, namely the full curl and the sum of the Murphy operators. We discuss their calculation also in terms of the framing factor associated to the appropriate irreducible representation of the quantum group, SU(N)_q.

We show that the theorem by Hemmer and Nakano, on uniqueness of Specht filtration multiplicities, can be proved working entirely with representations of symmetric groups, or Hecke algebras. Furthermore, we give a new proof that Schur algebras are quasi-hereditary provided the characteristic of the field is at least 5. Our tools are some more general results on stratifying systems.

In this article, we explicitly calculated the values of the representations of the Hecke algebra , associated with a Gelfand–Graev character of GL₄(q), at some of the standard basis elements.

We study the Schur elements and the a-function for cyclotomic Hecke algebras. As a consequence, we show the existence of canonical basic sets, as defined by Geck–Rouquier, for certain complex reflection groups. This includes the case of finite Weyl groups for all choices of parameters (in characteristic 0).

We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting monoidal categories, which have a geometric description via quiver varieties, in certain cases. We present basic properties of 2-representations and describe simple 2-representations, via cyclotomic quiver Hecke algebras, and through microlocalized quiver varieties.

Let p be a prime and B be a quaternion algebra indefinite over Q and ramified at p. We consider the space of quaternionic modular forms of weight k and level p^∞, endowed with the action of Hecke operators. By using cohomological methods, we show that the p-adic topological Hecke algebra does not depend on the weight k. This result provides a quaternionic version of a theorem proved by Hida for classical modular forms; we discuss the relationship of our result to Hida's theorem in terms of Jacquet–Langlands correspondence.

Let $({𝕋}_f,{𝔪}_f)$ denote the mod $p$ local Hecke algebra attached to a normalized Hecke eigenform $f$, which is a commutative algebra over some finite field ${𝔽}_q$ of characteristic $p$ and with residue field ${𝔽}_q$. By a result of Carayol we know that, if the residual Galois representation ${\overline{\rho }}_f:G_{\mathbb{Q}}\to {\mathrm{\text{GL}}}_2({𝔽}_q)$ is absolutely irreducible, then one can attach to this algebra a Galois representation ${\rho }_f:G_{\mathbb{Q}}\to {\mathrm{\text{GL}}}_2({𝕋}_f)$ that is a lift of ${\overline{\rho }}_f$. We will show how one can determine the image of ${\rho }_f$ under the assumptions that (i) the image of the residual representation contains ${\mathrm{\text{SL}}}_2({𝔽}_q)$, (ii) ${𝔪}_f^2=0$ and (iii) the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow to deduce the existence of certain $p$-elementary abelian extensions of big non-solvable number fields.

Let ${\mathcal{𝒮}}_k({\mathrm{\Gamma }}_0(N),\chi )$ denote the space of holomorphic cuspforms with Dirichlet character $\chi$ and modular subgroup ${\mathrm{\Gamma }}_0(N)$. We will characterize the space of newforms ${\mathcal{𝒮}}_k^{\mathrm{\text{new}}}({\mathrm{\Gamma }}_0(N),\chi )$ as the intersection of eigenspaces of a particular family of Hecke operators, generalizing previous work of Baruch–Purkait to forms with nontrivial character. We achieve this by obtaining representation theoretic results in the p-adic case which we then de-adelize into relations of classical Hecke operators.