[Math] Decomposing tensor product of lie algebra representations

Question

I'm given a lie algebra representation $\pi$ of some semi-simple algebra and that it decomposes into a sum of irreducible representations.

What technique should I use to show the decomposition of $\pi \otimes \pi$ into irreducible represenations?

Any clues will be highly appreciated.

Thanks in advance.

Answer 1

One approach to the general problem of decomposing a tensor product of irreducible finite-dimensional representations (hence any finite-dimensional representations) into irreducibles is to use the theory of crystals. The crystal of a representation is a colored directed graph associated to that representation. There is a purely combinatorial algorithm for producing the tensor product of two crystals. Then the connected components of the crystal graph correspond to the irreducible representations you're looking for. For many simple Lie algebras, the crystals of the irreducible finite-dimensional representations are described very explicitly. For instance, for $\mathfrak{sl}_n$, they are in terms of semistandard tableaux. So finding the decomposition you seek becomes combinatorics.