  We study a family of semiample divisors on the moduli space $\bar{M}_{0,n}$
that come from the theory of conformal blocks for the Lie algebra $sl_n$ and
level 1. The divisors we study are invariant under the action of $S_n$ on
$\bar{M}_{0,n}$. We compute their classes and prove that they generate extremal
rays in the cone of symmetric nef divisors on $\bar{M}_{0,n}$. In particular,
these divisors define birational contractions of $\bar{M}_{0,n}$, which we show
factor through reduction morphisms to moduli spaces of weighted pointed curves
defined by Hassett.
