  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.
