Combinatorial Yang–Baxter maps arising from the tetrahedron equation

We survey the matrix product solutions of the Yang–Baxter equation recently obtained from the tetrahedron equation. They form a family of quantum R-matrices of generalized quantum groups interpolating the symmetric tensor representations of Uq(An−1(1)) and the antisymmetric tensor representations of U−q−1(An−1(1))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${U_{ - {q^{ - 1}}}}\left( {A_{n - 1}^{\left( 1 \right)}} \right)$$\end{document} . We show that at q = 0, they all reduce to the Yang–Baxter maps called combinatorial R-matrices and describe the latter by an explicit algorithm.