  For fixed compact connected Lie groups H \subseteq G, we provide a polynomial
time algorithm to compute the multiplicity of a given irreducible
representation of H in the restriction of an irreducible representation of G.
Our algorithm is based on a finite difference formula which makes the
multiplicities amenable to Barvinok's algorithm for counting integral points in
polytopes.
  The Kronecker coefficients of the symmetric group, which can be seen to be a
special case of such multiplicities, play an important role in the geometric
complexity theory approach to the P vs. NP problem. Whereas their computation
is known to be #P-hard for Young diagrams with an arbitrary number of rows, our
algorithm computes them in polynomial time if the number of rows is bounded. We
complement our work by showing that information on the asymptotic growth rates
of multiplicities in the coordinate rings of orbit closures does not directly
lead to new complexity-theoretic obstructions beyond what can be obtained from
the moment polytopes of the orbit closures. Non-asymptotic information on the
multiplicities, such as provided by our algorithm, may therefore be essential
in order to find obstructions in geometric complexity theory.
