Orthogonal polynomial duality and unitary symmetries of multi--species ASEP$(q,\boldsymbol{\theta})$ and higher--spin vertex models via $^*$--bialgebra structure of higher rank quantum groups
| Autor: | Franceschini, Chiara, Kuan, Jeffrey, Zhou, Zhengye |
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| Rok vydání: | 2022 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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). Comment: 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 |
| Databáze: | arXiv |
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