  Let $X$ be a smooth, pointed Riemann surface of genus zero, and $G$ a simple,
simply-connected complex algebraic group. Associated to a finite number of
weights of $G$ and a level is a vector space called the space of conformal
blocks, and a vector bundle of conformal blocks over $\bar{\text{M}}_{0,n}$. We
show that, assuming the weights are on a face of the multiplicative eigenvalue
polytope, the space of conformal blocks is isomorphic to a product of conformal
blocks over groups of lower rank. If the weights are on a degree zero wall,
then we also show that there is an isomorphism of conformal blocks bundles,
giving an explicit relation between the associated nef divisors. The methods of
the proof are geometric, and use the identification of conformal blocks with
spaces of generalized theta functions, and the moduli stacks of parahoric
bundles recently studied by Balaji and Seshadri.
