MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For SU2 and even SU2(q) the triangle condition is, well, the triangle condition (conveniently, all irreps are described by (half)integer J completely). Additionally, all three J of a triple must add to integer.

But what is the analogue to that for some arbitrary (quantum) group? As usual, I don't have a clue but an assumption :-) - random example in group A3: Let the reps be 1R1+2R2+1R3,2R1+R2+1R3 and 5R1+2R2+2R3, then you check every irrep separately: 125 - triangle fail, 212 - parity fail, 112 - pass, overall: fail.

Of course, I could be totally wrong...What is the correct condition? (Maybe it's not even a simple one: outer multiplicity and that.)

share|cite|improve this question
up vote 6 down vote accepted

As you probably guessed the answer is "its complicated." I guess you mean what is the rule for when the tensor product of two irreps $V_\lambda \otimes V_\gamma$ contains an irrep $V_\delta$. The answer involves identifying the irrep with its highest weight vector $\lambda$ and viewing it in the Weyl chamber. Humphreys "Intro to Lie Algebra and ..." is the a good place to learn this stuff. Roughly the analog of the triangle inequality says that if you add all images of $\lambda$ under the Weyl group to $\gamma$, $\delta$ will fall in the convex hull. the analog of the integrality condition is that $\lambda+\gamma - \delta$ must be in the root lattice: in general the weight lattice mod the root lattice is a small abelian group that Humphreys will tell you about. The full story for the tensor product decomposition is given by the beautiful Racah formula, which can be visualized in rank 2 thus: mark the dimensions of the weights of $V_\lambda$ in tracing paper in the weight lattice. Slide it to put the 0 over $\gamma$. Fold the tracing paper along the walls of the extended Weyl chamber until it lies entirely in that chamber. add the numbers over $\delta$, with minus signs on the ones written backwards (i.e. an odd number of folds). That is the multiplicity of $V_\delta$.

Similar story holds for quantum groups at roots of unity (you have to throw away pesky nonirreducible reps), and there is a perfectly analogous quantum Racah formula with the Weyl alcove taking the role of the chamber. To toot my own horn, see on archive for a pretty full treatment.

share|cite|improve this answer
Too bad it's not as simple as SU2. BTW, I read about Weyl chambers before, but I always thought they contained instruments of math torture designed for amateurs like me :-) – Hauke Reddmann Oct 5 '11 at 9:31

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.