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arXiv · 2212.06404

Solving the n-color ice model

Patrick Addona·Ethan Bockenhauer·Ben Brubaker·Michael Cauthorn·Cianan Conefrey-Shinozaki·David Donze·William Dudarov·Jessamyn Dukes·Andrew Hardt·Cindy Li·Jigang Li·Yanli Liu·Neelima Puthanveetil·Zain Qudsi·Jordan Simons·Joseph Sullivan·Autumn Young
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Abstract

Given an arbitrary choice of two sets of nonzero Boltzmann weights for $n$-color lattice models, we provide explicit algebraic conditions on these Boltzmann weights which guarantee a solution (i.e., a third set of weights) to the Yang-Baxter equation. Furthermore we provide an explicit one-dimensional parametrization of all solutions in this case. These $n$-color lattice models are so named because their admissible vertices have adjacent edges labeled by one of $n$ colors with additional restrictions. The two-colored case specializes to the six-vertex model, in which case our results recover the familiar quadric condition of Baxter for solvability. The general $n$-color case includes important solutions to the Yang-Baxter equation like the evaluation modules for the quantum affine Lie algebra $U_q(\hat{\mathfrak{sl}}_n)$. Finally, we demonstrate the invariance of this class of solutions under natural transformations, including those associated with Drinfeld twisting.

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Patrick Addona, Ethan Bockenhauer, Ben Brubaker, Michael Cauthorn, Cianan Conefrey-Shinozaki, David Donze, William Dudarov, Jessamyn Dukes, Andrew Hardt, Cindy Li, Jigang Li, Yanli Liu, Neelima Puthanveetil, Zain Qudsi, Jordan Simons, Joseph Sullivan, Autumn Young. 2022-12-13. Solving the n-color ice model. https://doi.org/10.1007/s00026-025-00770-1

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