On the Weights of Root Subgroups of Affine Toric Varieties

Let $X$ be an affine toric variety and let $D(X)$ be the set of weights of all root subgroups. It is known that $D(X)$ together with its embedding into the character group determines $X$ as a toric variety. In this article we prove that $X$ is already determined by the abstract set $D(X)$ together with some additional combinatorial data.