The two-boundary Temperley–Lieb algebra

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 N2 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.