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

Abstract

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 \textit{unitary} symmetries, which imply the orthogonality of the duality functions. In the case of the quantum group ${\mathcal{U}}_q({\mathfrak{gl}}_{n+1})$, the result is a nested multivariate $q$--Krawtchouk duality for the $n$--species ASEP$(q,\mathbf{\theta })$. 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).

Version 3 is the journal version. For an accessible version of the PDF, please visit the second author's website at http://math.tamu.edu/~jkuan/arXiv-2209.03531.html

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