Orthogonal Polynomial Duality and Unitary Symmetries of Multi-species ASEP\((q,\varvec{\theta })\) and Higher-Spin Vertex Models via \(^*\)-Bialgebra Structure of Higher Rank Quantum Groups

We propose a general method to produce orthogonal polynomial dualities from the \(^*\) ${}^{\ast }$ -bialgebra structure of Drinfeld–Jimbo quantum groups. The \(^*\) ${}^{\ast }$ -structure allows for the construction of certain unitary symmetries, which imply the orthogonality of the duality functions. In the case of the quantum group \(\mathcal {U}_q(\mathfrak {gl}_{n+1})\) ${\mathcal{U}}_q({\mathfrak{gl}}_{n+1})$ , the result is a nested multivariate q-Krawtchouk duality for the n-species ASEP \((q,\varvec{\theta }) \) $(q,\mathit{\theta })$ . The method also applies to other quantized simple Lie algebras and to stochastic vertex models. As a probabilistic application of the duality relation found, we provide the explicit formula of the q-shifted factorial moments (namely the q-analogue of the Pochhammer symbol) for the two-species q-TAZRP (totally asymmetric zero range process).