On $S_n$-invariant conformal blocks vector bundles of rank one on ${\bar{M}}_{0,n}$

For any simple Lie algebra, a positive integer, and tuple of compatible weights, the conformal blocks bundle is a globally generated vector bundle on the moduli space of pointed rational curves. We classify all $S_n$-invariant vector bundles of conformal blocks for ${\mathfrak{s}\mathfrak{l}}_n$ which have rank one. We show that the cone generated by their base point free first Chern classes is polyhedral, generated by level one divisors.