Semigroups of $s{l}_3(\mathbb{C})$ tensor product invariants

We compute presentations for a family of semigroup algebras related to the problem of decomposing $s{l}_3(\mathbb{C})$ tensor products. Along the way we find new toric degenerations of the Grassmannian variety $G{r}_3({\mathbb{C}}^n)$ which $T-$invariant for $T\subset G{L}_n(\mathbb{C})$ the diagonal torus.