Some Algebraic Aspects of the Inhomogeneous Six-Vertex Model

The inhomogeneous six-vertex model is a $2D$ multiparametric integrable statistical system. In the scaling limit it is expected to cover different classes of critical behaviour which, for the most part, have remained unexplored. For general values of the parameters and twisted boundary conditions the model possesses $\mathrm{U}(1)$ invariance. In this paper we discuss the restrictions imposed on the parameters for which additional global symmetries arise that are consistent with the integrable structure. These include the lattice counterparts of ${\mathcal C}$, ${\mathcal P}$ and ${\mathcal T}$ as well as translational invariance. The special properties of the lattice system that possesses an additional ${\mathcal Z}_r$ invariance are considered. We also describe the Hermitian structures, which are consistent with the integrable one. The analysis lays the groundwork for studying the scaling limit of the inhomogeneous six-vertex model.