A short note on Markov duality in multi–species higher spin stochastic vertex models

Abstract

We show that the multi–species higher spin stochastic vertex model, also called the $U_q(A_n^{(1)})$ vertex model, satisfies a duality where the indicator function has the form $\{{\mathrm{\eta }}_{[i,n]}^x\ge {\mathrm{\xi }}_{[i,n]}^x\}$. In other words, for every particle in the ξ configuration of species i at vertex x, there must be a particle of species $j\ge i$ at vertex x in the η configuration. For these duality functions, the dual process has fewer particles than the original process, making it suitable for applications. The proof follows by applying charge reversal to previously discovered duality functions, which also results in open boundary conditions. As a special case, we recover the duality for the stochastic six vertex model recently found by Y. Lin.