Finite-dimensional representations of a shock algebra

The algebra describing a shock measure in the asymmetric simple exclusion model, seen from a second class particle, has finite-dimensional representations if and only if the asymmetry parameter p of the model and the left and right asymptotic densities rho(+/-) of the shock satisfy [(1-p)/p](r)=rho(-)(1-rho(+))/rho(+)(1-rho(-)) for some integer r greater than or equal to 1; the minimal dimension of the representation is then 2r. These representations can be used to calculate correlation functions in the model.