  We study a two-boundary extension of the Temperley-Lieb algebra which has
recently arisen in statistical mechanics. This algebra lies in a quotient of
the affine Hecke algebra of type C and has a natural diagrammatic
representation. The algebra has three parameters and, for generic values of
these, we determine its representation theory.
  We use the action of the centre of the affine Hecke algebra to show that all
irreducible representations lie within a finite dimensional diagrammatic
quotient. These representations are fully characterised by an additional
parameter related to the action of the centre. For generic values of this
parameter there is a unique representation of dimension 2^N and we show that it
is isomorphic to a tensor space representation. We construct a basis in which
the Gram matrix is diagonal and use this to discuss the irreducibility of this
representation.
