  We consider a conjecture that identifies two types of base point free
divisors on $\bar{M}_{0,n}$. The first arises from Gromov-Witten theory of a
Grassmannian. The second comes from first Chern classes of vector bundles
associated to simple Lie algebras in type A. Here we reduce this conjecture on
$\bar{M}_{0,n}$ to the same statement for $n=4$. A reinterpretation leads to a
proof of the conjecture on $\bar{M}_{0,n}$ for a large class, and we give
sufficient conditions for the non-vanishing of these divisors.
