Orthogonal polynomial duality and unitary symmetries of multi--species ASEP$(q,\mathit{\theta })$ and higher--spin vertex models via ${}^{\ast }$--bialgebra structure of higher rank quantum groups

We propose a novel, 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 $\mathit{\text{unitary}}$ symmetries, which imply the orthogonality of the duality functions. In the case of the quantum group ${\mathcal{U}}_q({\mathfrak{g}\mathfrak{l}}_{n+1})$, the result is a nested multivariate $q$–Krawtchouk duality for the $n$–species ASEP$(q,\mathit{\theta })$. 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).