Matrix product solutions to the $G_2$ reflection equation

TL;DR

This paper investigates the $G_2$ reflection equation, a complex scattering relation in 1+1 dimensions, and constructs matrix product solutions using representation theory of the quantized coordinate ring $A_q(G_2)$.

Contribution

It introduces matrix product solutions to the $G_2$ reflection equation, extending the framework of integrable models to the exceptional Lie group $G_2$.

Findings

Established a connection between the $G_2$ reflection equation and $A_q(G_2)$ representation theory.

Constructed explicit matrix product solutions for the $G_2$ reflection equation.

Extended the integrable systems framework to include $G_2$-type relations.

Abstract

We study the $G_2$ reflection equation for the three particles in $1+1$ dimension that undergo a special scattering/reflections described by the Pappus theorem. It is a sixth order equation and serves as a natural $G_2$ analogue of the Yang-Baxter and the reflection equations corresponding to the cubic and the quartic Coxeter relations of type $A$ and $BC$, respectively. We construct matrix product solutions to the $G_2$ reflection equation by exploiting a connection to the representation theory of the quantized coordinate ring $A_q(G_2)$.