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A recent paper \cite{KMMO} introduced the stochastic U_q(A_n^{(1)}) vertexmodel. The stochastic S-matrix is related to the R-matrix of the quantum groupU_q(A_n^{(1)}) by a gauge transformation. We will show that a certain functionD^+_{\mu} intertwines with the transfer matrix and its space reversal. Wheninterpreting the transfer matrix as the transition matrix of a discrete-timetotally asymmetric particle system on the one-dimensional lattice Z, thefunction D^+_{\mu} becomes a Markov duality function D_{\mu} which only dependson q and the vertical spin parameters \mu_x. By considering degenerations inthe spectral parameter, the duality results also hold on a finite lattice withclosed boundary conditions, and for a continuous-time degeneration. Thisduality function had previously appeared in a multi-species ASEP(q,j) process.The proof here uses that the R-matrix intertwines with the co-product, but doesnot explicitly use the Yang-Baxter equation.  It will also be shown that the stochastic U_q(A_n^{(1)}) is a multi-speciesversion of a stochastic vertex model studied in \cite{BP,CP}. This will be doneby generalizing the fusion process of \cite{CP} and showing that it matches thefusion of \cite{KRL} up to the gauge transformation.  We also show, by direct computation, that the multi-species q-Hahn Bosonprocess (which arises at a special value of the spectral parameter) alsosatisfies duality with respect to D_0, generalizing the single-species resultof \cite{C}.

An algebraic construction of duality functions for the stochastic U_q(A_n^{(1)}) vertex model and its degenerations