Fused braids and centralisers of tensor representations of $U_q(gl_N)$

TL;DR

This paper introduces fused permutation and fused Hecke algebras to establish a Schur--Weyl duality for tensor products of symmetrised powers of $U_q(gl_N)$'s natural representation, and describes their irreducible representations and centralisers.

Contribution

It defines fused algebras and proves a Schur--Weyl duality for tensor products of symmetrised powers of $U_q(gl_N)$, including explicit descriptions of irreducible representations and centralisers.

Findings

Established Schur--Weyl duality for fused algebras and tensor products.

Explicit descriptions of irreducible representations and branching rules.

Conjectured and partially proved generators for centralisers.

Abstract

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 $U_q(sl_2)$.