On S_n-invariant conformal blocks vector bundles of rank one on overline 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 $\mathfrak{sl}_n$ 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.