  We give an explicit minimal graded free resolution, in terms of
representations of the symmetric group $S_d$, of a Galois-theoretic
configuration of $d$ points in $\mathbb{P}^{d-2}$ that was studied by Bhargava
in the context of ring parametrizations. When applied to the geometric generic
fiber of a simply branched degree $d$ cover of $\mathbb{P}^1$ by a relatively
canonically embedded curve $C$, our construction gives a new interpretation for
the splitting types of the syzygy bundles appearing in its relative minimal
resolution. Concretely, our work implies that all these splitting types consist
of scrollar invariants of resolvent covers. This vastly generalizes a prior
observation due to Casnati, namely that the first syzygy bundle of a degree $4$
cover splits according to the scrollar invariants of its cubic resolvent. Our
work also shows that the splitting types of the syzygy bundles, together with
the multi-set of scrollar invariants, belong to a much larger class of
multi-sets of invariants that can be attached to $C \to \mathbb{P}^1$: one for
each irreducible representation of $S_d$, i.e., one for each partition of $d$.
