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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.

user71158
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  • Do you just need to show that it decomposes, or do you need to find the summands and their multiplicities? For the latter, you can reduce it to figuring out how the tensor product of two arbitrary simple modules splits (which can actually be fairly complicated). – Tobias Kildetoft Apr 16 '13 at 14:03
  • I think I need to show the later. If it's the standard representation we are talking about, is it any easier? – user71158 Apr 16 '13 at 15:01
  • Well, the tensor product of the standard representation with itself has two submodules that are "easy" to find, namely the second symmetric power and the second exterior power. – Tobias Kildetoft Apr 16 '13 at 17:22
  • Right! Somehow I missed that one. But then, what is the significance of the fact that $\pi$ decomposes into sum of irreducible representations? Is it some precondition? – user71158 Apr 16 '13 at 18:26
  • Any finite dimensional representation of a semisimple Lie algebra decomposes into a direct sum of irreducibles, so that is not really extra information. – Tobias Kildetoft Apr 16 '13 at 18:28
  • There is a product formula for product of two irreducible representations. This is standard. Just find it out in a standard textbook. – daniel Mar 23 '17 at 03:32

2 Answers2

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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.

Alistair Savage
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Depending on the sizes of the representations involved, you may be able to solve your decomposition problem using the Weyl character formula. This is especially true if your representations are finite-dimensional. If so, then the character is a complete invariant of the representation, and it is additive and multiplicative under direct sum and tensor product.

The Weyl character formula tells you the character of an irrep in terms of its highest weight. Conversely, given any character, you can find the highest weight with nonzero coefficient; subtract off the corresponding character, and repeat. In practice, this is pretty fast by hand for $\mathrm{sl}(3)$, and on a computer for larger things.

For more complicated problems, I've heard very good things about the computer algebra package LiE, although I have never used it myself.

Theo Johnson-Freyd
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