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Motivated by a study of the crossing symmetry of the `gemini' representationof the affine Hecke algebra we give a construction for crossing tensor spacerepresentations of ordinary Hecke algebras. These representations buildsolutions to the Yang--Baxter equation satisfying the crossing condition (thatis, integrable quantum spin chains). We show that every crossing representationof the Temperley--Lieb algebra appears in this construction, and in particularthat this construction builds new representations. We extend these to newrepresentations of the blob algebra, which build new solutions to the BoundaryYang--Baxter equation (i.e. open spin chains with integrable boundaryconditions).  We prove that the open spin chain Hamiltonian derived from Sklyanin'scommuting transfer matrix using such a solution can always be expressed as therepresentation of an element of the blob algebra, and determine this element.We determine the representation theory (irreducible content) of the newrepresentations and hence show that all such Hamiltonians have the samespectrum up to multiplicity, for any given value of the algebraic boundaryparameter. (A corollary is that our models have the same spectrum as the openXXZ chain with nondiagonal boundary -- despite differing from this model inhaving reference states.) Using this multiplicity data, and other ideas, weinvestigate the underlying quantum group symmetry of the new Hamiltonians. Wederive the form of the spectrum and the Bethe ansatz equations.Comment: 43 pages, multiple figure

On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary