Invariant rings of sums of fundamental representations of SLn and colored hypergraphs

Abstract

The fundamental representations of the special linear group SLn over the complex numbers are the exterior powers of Cn. We consider the invariant rings of sums of arbitrarily many copies of these SLn-modules. We use the symbolic method for antisymmetric tensors developed by Grosshans, Rota and Stein, 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 SL4 and SL5, as well as syzygies for SL4. Since the invariants constitute incidence geometry of linear subspaces of the projective space Pn−1, the generating invariants provide (minimal) sets of geometric relations that are able to describe all others.