Given a representation $R$ of some group $G$ one can find in many books and papers (e.g. page 96ff here) the decomposition under certain subgroups:
$$ R= R_1 + R_2 + \ldots$$
This is often called a branching rule.
For example, for $ SU(4) \subset SU(5) $ we have the decomposition
$$ 15=1+4+10 $$
This can then be used to find a map, that maps the weights of the $15$ rep of $SU(5)$ to the corresponding weights of the $SU(4)$ reps. Commonly this map is written in terms of a projection matrix. Nevertheless, in my understanding, this can only be done once we know the decomposition of some representations under the subgroup in question.
How does one determine a decomposition like $15=1+4+10$, without a given projection matrix?