Invariant rings of sums of fundamental representations of ${\rm SL}_n$ and colored hypergraphs

Abstract

The fundamental representations of the special linear group ${\rm SL}_n$ over the complex numbers are the exterior powers of $\mathbb{C}^n$. We consider the invariant rings of sums of arbitrary many copies of these ${\rm SL}_n$-modules. The symbolic method for antisymmetric tensors developed by Grosshans, Rota and Stein is used, but instead of brackets, we associate colored hypergraphs to the invariants. This approach allows us to use results and insights from graph theory. In particular, we determine (minimal) generating sets of the invariant rings in the case of ${\rm SL}_4$ and ${\rm SL}_5$, as well as syzygies for ${\rm SL}_4$. Since the invariants constitute incidence geometry of linear subspaces of the projective space $\mathbb{P}_{n-1}$, the generating invariants provide (minimal) sets of geometric relations that are able to describe all others.