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The fundamental representations of the special linear group ${\rm SL}_n$ overthe complex numbers are the exterior powers of $\mathbb{C}^n$. We consider theinvariant rings of sums of arbitrary many copies of these ${\rm SL}_n$-modules.The symbolic method for antisymmetric tensors developed by Grosshans, Rota andStein is used, but instead of brackets, we associate colored hypergraphs to theinvariants. This approach allows us to use results and insights from graphtheory. In particular, we determine (minimal) generating sets of the invariantrings 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 linearsubspaces of the projective space $\mathbb{P}_{n-1}$, the generating invariantsprovide (minimal) sets of geometric relations that are able to describe allothers.

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