  Following the methods used by Derksen-Weyman in \cite{DW11} and Chindris in
\cite{Chi08}, we use quiver theory to represent the generalized
Littlewood-Richardson coefficients for the branching rule for the diagonal
embedding of $\gl(n)$ as the dimension of a weight space of semi-invariants.
Using this, we prove their saturation and investigate when they are nonzero. We
also show that for certain partitions the associated stretched polynomials
satisfy the same conjectures as single Littlewood-Richardson coefficients. We
then provide a polytopal description of this multiplicity and show that its
positivity may be computed in strongly polynomial time. Finally, we remark that
similar results hold for certain other generalized Littlewood-Richardson
coefficients.
