Mathematics > Probability
[Submitted on 8 Sep 2022 (v1), last revised 10 Mar 2024 (this version, v3)]
Title:Orthogonal polynomial duality and unitary symmetries of multi--species ASEP$(q,\boldsymbolθ)$ and higher--spin vertex models via $^*$--bialgebra structure of higher rank quantum groups
View PDF HTML (experimental)Abstract:We propose a novel, general method to produce orthogonal polynomial dualities from the $^*$--bialgebra structure of Drinfeld--Jimbo quantum groups. The $^*$--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$(q,\boldsymbol{\theta})$. 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).
Submission history
From: Jeffrey Kuan [view email][v1] Thu, 8 Sep 2022 02:14:10 UTC (177 KB)
[v2] Mon, 10 Oct 2022 15:20:00 UTC (179 KB)
[v3] Sun, 10 Mar 2024 05:55:52 UTC (192 KB)
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