Verma Modules over Restricted Quantum $\mathfrak{sl}_3$ at a Fourth Root of Unity

For a semisimple Lie algebra $\mathfrak{g}$ of rank $n$, let $\overline{U}_\zeta(\mathfrak{g})$ be the restricted quantum group of $\mathfrak{g}$ at a primitive fourth root of unity. This quantum group admits a natural Borel-induced representation $V(\underline{\boldsymbol{t}})$, with $\underline{\boldsymbol{t}}\in(\mathbb{C}^\times)^n$ determined by a character on the Cartan subalgebra. Ohtsuki showed that for $\mathfrak{g}=\mathfrak{sl}_2$, the braid group representation determined by tensor powers of $V(t)$ is the exterior algebra of the Burau representation. In this paper, we begin work on the $\mathfrak{g}=\mathfrak{sl}_3$ case. This includes a generalization of the decomposition for $V(\underline{\boldsymbol{t}})\otimes V(\underline{\boldsymbol{s}})$ used in Ohtsuki's work, which we expect to hold for any $\mathfrak{g}$. We also define a stratification of $(\mathbb{C}^\times)^{4}$ whose points $(\underline{\boldsymbol{t}},\underline{\boldsymbol{s}})$ in the lower strata are associated to representations $V(\underline{\boldsymbol{t}})\otimes V(\underline{\boldsymbol{s}})$ which do not have a homogeneous cyclic generator. Moreover, we characterize exactly when the isomorphism ${V(\underline{\boldsymbol{t}})\otimes V(\underline{\boldsymbol{s}})\cong V(\underline{\boldsymbol{\lambda t}})\otimes V(\underline{\boldsymbol{\lambda^{-1} s}})}$ holds.