Semigroups of $sl_3(\mathbb{C})$ tensor product invariants

We compute presentations for a family of semigroup algebras related to the problem of decomposing $sl_3(\mathbb{C})$ tensor products. Along the way we find new toric degenerations of the Grassmannian variety $Gr_3(\mathbb{C}^n)$ which $T-$invariant for $T \subset GL_n(\mathbb{C})$ the diagonal torus.