Combinatorial Yang-Baxter maps arising from tetrahedron equation

Abstract

We survey the matrix product solutions of the Yang-Baxter equation obtained recently from the tetrahedron equation. They form a family of quantum $R$ matrices of generalized quantum groups interpolating the symmetric tensor representations of $U_q(A^{(1)}_{n-1})$ and the anti-symmetric tensor representations of $U_{-q^{-1}}(A^{(1)}_{n-1})$. We show that at $q=0$ they all reduce to the Yang-Baxter maps called combinatorial $R$, and describe the latter by explicit algorithm.