  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.
