Generalized matrix Ansatz in the multispecies exclusion process-the partially asymmetric case

We investigate one of the simplest multispecies generalizations of the asymmetric simple exclusion process on a ring. This process has a rich combinatorial spectral structure and a matrix product form for the stationary state. In the totally asymmetric case, operators that conjugate the dynamics of systems with different numbers of species were obtained by the authors and recently reported by Arita et al (2011 J. Phys. A: Math. Theor. 44 335004). The existence of such nontrivial operators was reformulated as a representation problem for a specific quadratic algebra (generalized matrix Ansatz). In this work, we construct the family of representations explicitly for the partially asymmetric case. This solution cannot be obtained by a simple deformation of the totally asymmetric case.