Fused Hecke algebra and one-boundary algebras

This paper gives an algebraic presentation of the fused Hecke algebra which describes the centraliser of tensor products of the U q (gl N )-representation labelled by a one-row partition of any size with vector representations.It is obtained through a detailed study of a new algebra that we call the symmetric one-boundary Hecke algebra.In particular, we prove that the symmetric one-boundary Hecke algebra is free over a ring of Laurent polynomials in three variables and we provide a basis indexed by a certain subset of signed permutations.We show how the symmetric one-boundary Hecke algebra admits the one-boundary Temperley-Lieb algebra as a quotient, and we also describe a basis of this latter algebra combinatorially in terms of signed permutations with avoiding patterns.The quotients corresponding to any value of N in gl N (the Temperley-Lieb one corresponds to N = 2) are also introduced.Finally, we obtain the fused Hecke algebra, and in turn the centralisers for any value of N, by specialising and quotienting the symmetric one-boundary Hecke algebra.In particular, this generalises to the Hecke case the description of the so-called boundary seam algebra, which is then obtained (taking N = 2) as a quotient of the fused Hecke algebra.