Let \(C\) be a smooth curve of genus \(g\) and \(f: C\to \mathbb{P}^1\) a degree \(d\) covering, both defined over a field \(k\) with either characteristic \(0\) or characteristic \(>d\). The scrollar invariants of \(f\) are the \(d-1\) integers \(e_1\le \cdots \le e_{d-1}\) such that \(f_\ast(\mathcal {O}_C)\cong \mathcal {O}_{\mathbb {P}^1}\oplus \mathcal {O}_{\mathbb {P}^1}(-e_1)\oplus \cdots \oplus \mathcal {O}_{\mathbb {P}^1}(-e_{d-1})\). Now assume that \(f\) is simply branched. With this assumption the Galois group of \(f\) is the full symmetric group \(S_d\). Let \(f': C'\to \mathbb {P}^1\) be the associated Galois covering. The indecomposable representations of \(S_d\) are described by the partitions of \(d\). The function field \(k(C')\) is seen as an \(S_d\)-representation. The vector bundle \(f'_\ast (\mathcal {O}_{C'})\) is a direct sum of vector bundles, one for each partition \(\lambda\) of \(d\), and each of these factors is a direct sum of line bundles.
In this way the authors partitions the scrollar invariant of \(f'\) in term of the representations. The first main theorem of the papers described all the syzygy bundles of the relative canonical resolution of \(C\) with respect to \(f\) (introduced in [G. Casnati and T. Ekedahl, J. Algebr. Geom. 5, No. 3, 439–460 (1996; Zbl 0866.14009)] in term of scrollar invariants of \(f'\). The paper contains several other gems, very useful for covering of curves, e.g. the genus of \(C'\) and new restrictions on the scrollar invariants.They get invariants also with respect to subgroup \(H\) of \(S_d\). Among the problems arised: under what conditions on \(d\), \(g\) and the partition \(\lambda\) of \(d\) are connedted the scrollar invariants of the factor associated to \(\lambda\)?