Combinatorial Yang-Baxter maps arising from tetrahedron equation

TL;DR

This paper surveys matrix product solutions to the Yang-Baxter equation derived from the tetrahedron equation, revealing a family of quantum R-matrices that interpolate between different tensor representations and reduce to combinatorial R-maps at q=0.

Contribution

It introduces a family of quantum R-matrices from the tetrahedron equation that interpolate between symmetric and anti-symmetric tensor representations, and explicitly describes the combinatorial R-maps at q=0.

Findings

Family of quantum R-matrices interpolates between tensor representations.

At q=0, all solutions reduce to combinatorial R-maps.

Explicit algorithm for combinatorial R-maps provided.

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.