I would like to add to the answers by Ben and Allen. First if we extend the question to include all multiplicities and not just the multiplicity of the trivial representation then there are a number of special cases that are of interest:
1.Take $H$ to be the trivial group then the question asks for the dimension of a representation.
2.Take $H$ to be a maximal torus then we are asking for the character of a representation.
3. Take $G=Htimes H$ and $H$ the diagonal subgroup. Then we are asking for tensor product multiplicities.
4. For $V$ a representation of $K$. Take $G=SL(V)$ and $H=K$. Then we are calculating plethysms.
A paper that discusses this which gives a formula for branching rules is:
MR1120029 (92f:22022) Cohen, Arjeh M. ; Ruitenburg, G. C. M. Generating functions and Lie groups.
Computational aspects of Lie group representations and related topics
(Amsterdam, 1990),
19--28, CWI Tract, 84, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1991.
As I understand it both Ben and Allen agree that this is not a simple way of finding branching rules. The reason is that this involves a sum over the Weyl group.
If you take the special cases above then historically the first solutions to these problems were given by formulae involving a sum over the Weyl group. For some of these special cases there are solutions which don't involve cancelling terms. For example, LiE
calculates these without summing over the Weyl group. The LiE home page is
http://www-math.univ-poitiers.fr/~maavl/LiE/index.html
and the LiE manual does describe how these special cases are implemented.
However LiE treats each of these special cases separately. I think it is an interesting question whether there is an algorithm for finding branching rules which could be implemented in LiE and which does not involve a sum over the Weyl group.
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