We propose a general method to produce orthogonal polynomial dualities from the \(^*\)-bialgebra structure of Drinfeld–Jimbo quantum groups. The \(^*\)-structure allows for the construction of certain unitary symmetries, which imply the orthogonality of the duality functions. In the case of the quantum group \(\mathcal {U}_q(\mathfrak {gl}_{n+1})\), the result is a nested multivariate q-Krawtchouk duality for the n-species ASEP\((q,\varvec{\theta }) \). 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).

我们提出了一种从Drinfeld-Jimbo 量子群的\(^*\) -双代数结构产生正交多项式对偶性的通用方法。 \(^*\)结构允许构造某些酉对称性，这意味着对偶函数的正交性。在量子群\(\mathcal {U}_q(\mathfrak {gl}_{n+1})\)的情况下，结果是n种 ASEP的嵌套多元q -Krawtchouk 对偶性\(( q,\varvec{\theta }) \)。该方法也适用于其他量化的简单李代数和随机顶点模型。作为所发现的对偶关系的概率应用，我们提供了双种类q -TAZRP（完全不对称零范围过程） 的q平移阶乘矩（即Pochhammer 符号的q类似物）的显式公式。