(This might get more traction on math.stackexchange.com, rather than physics.stackexchange.com)

One can easily answer some of these questions with the concept of the character of a representation.

A (linear) representation is a homomorphism from group elements to linear transformations of a vector space. The character of a representation is a map from group elements to the trace of this linear transformation. A representation that is a direct sum of other representations has a character that is the ordinary sum of the characters of these other representations.

One can define an inner product on characters $x$ and $y$ by doing a normalized sum or integral of $\overline{x(g)} y(g)$. Irreducible characters are orthonormal. This lets you test for irreducibility -- the character of the representation must have an inner product of 1 with itself. If you have a list of the irreducible characters (which may be easier to construct than the list of irreducible representations), you can use this inner product to count how many times each irreducible character appears in a given character (and hence how many times an irreducible representation appears in a representation).

Of course, you do need the list of irreducible characters and representations to do this. For finite groups, simple counting arguments plus orthogonality can get you an exhaustive list of irreducible characters, which will then tell you quite a bit about the corresponding representations. For continuous groups, other techniques are required.

This post imported from StackExchange Mathematics at 2014-06-09 19:10 (UCT), posted by SE-user wnoise