  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.
