Matrix product solutions to the $G_2$ reflection equation

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 $A_q(G_2)$.