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An Algebraic Construction of Duality Functions for the Stochastic \({\mathcal{U}_q( A_n^{(1)})}\) Vertex Model and Its Degenerations

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Abstract

A recent paper (Kuniba in Nucl Phys B 913:248–277, 2016) introduced the stochastic \({\mathcal{U}_q(A_n^{(1)})}\) vertex model. The stochastic S-matrix is related to the R-matrix of the quantum group \({\mathcal{U}_q(A_n^{(1)})}\) by a gauge transformation. We will show that a certain function \({D^+_{\vec{m}}}\) intertwines with the transfer matrix and its space reversal. When interpreting the transfer matrix as the transition matrix of a discrete-time totally asymmetric particle system on the one-dimensional lattice \({\mathbb{Z}}\), the function \({D^+_{\vec{m}}}\) becomes a Markov duality function \({D_{\vec{m}}}\) which only depends on q and the vertical spin parameters \({\{\mu_x\}}\). By considering degenerations in the spectral parameter, the duality results also hold on a finite lattice with closed boundary conditions, and for a continuous-time degeneration. This duality function had previously appeared in a multi-species ASEP(q, j) process (Kuan in A multi-species ASEP(q, j) and q-TAZRP with stochastic duality, 2017). The proof here uses that the R-matrix intertwines with the co-product, but does not explicitly use the Yang–Baxter equation. It will also be shown that the stochastic \({\mathcal{U}_q(A_n^{(1)})}\) is a multi-species version of a stochastic vertex model studied in Borodin and Petrov (Higher spin six vertex model and symmetric rational functions, 2016) and Corwin and Petrov (Commun Math Phys 343:651–700, 2016). This will be done by generalizing the fusion process of Corwin and Petrov (2016) and showing that it matches the fusion of Kulish and yu (Lett Math Phys 5:393–403, 1981) up to the gauge transformation. We also show, by direct computation, that the multi-species q-Hahn Boson process (which arises at a special value of the spectral parameter) also satisfies duality with respect to \({D_{\infty}}\), generalizing the single-species result of Corwin (Int Math Res Not 2015:5577–5603, 2015).

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  1. Columbia University, New York, NY, USA

    Jeffrey Kuan

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Communicated by A. Borodin

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Kuan, J. An Algebraic Construction of Duality Functions for the Stochastic \({\mathcal{U}_q( A_n^{(1)})}\) Vertex Model and Its Degenerations. Commun. Math. Phys. 359, 121–187 (2018). https://doi.org/10.1007/s00220-018-3108-x

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