We present in this paper the algebra of fused permutations and its deformation the fused Hecke algebra. The first one is defined on a set of combinatorial objects that we call fused permutations, and its deformation is defined on a set of topological objects that we call fused braids. We use these algebras to prove a Schur-Weyl duality theorem for any tensor products of any symmetrised powers of the natural representation of U q (gl N ). Then we proceed to the study of the fused Hecke algebras and in particular, we describe explicitely the irreducible representations and the branching rules. Finally, we aim to an algebraic description of the centralisers of the tensor products of U q (gl N )-representations under consideration. We exhibit a simple explicit element that we conjecture to generate the kernel from the fused Hecke algebra to the centraliser. We prove this conjecture in some cases and in particular, we obtain a description of the centraliser of any tensor products of any finite-dimensional representations of U q (sl 2 ).
We present explicit formulas for the operators providing missing labels for the tensor product of two irreducible representations of su 3 . The result is seen as a particular representation of the diagonal centraliser of su 3 through a pair of tridiagonal matrices. Using these explicit formulas, we investigate the symmetry of this missing label problem and we find a symmetry group of order 144 larger than what can be expected from the natural symmetries. Several realisations of this symmetry group are given, including an interpretation as a subgroup of the Weyl group of type E 6 , which appeared in an earlier work as the symmetry group of the diagonal centraliser. Using the combinatorics of the root system of type E 6 , we provide a family of representations of the diagonal centraliser by infinite tridiagonal matrices, from which all the finite-dimensional representations affording the missing label can be extracted. Besides, some connections with the Hahn algebra, Heun-Hahn operators and Bethe ansatz are discussed along with some similarities with the wellknown symmetries of the Clebsch-Gordan coefficients.
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