Question

Direct product of spin-s representations can be decomposed into direct sum of some other spin-s representation:

{j} ⊗ {s} = ∑ ⊕l = |j−s| j+s {l}
Howard Georgi gives a hint in his book Lie Algebras in Particle Physics that this can be proved using the highest weight decomposition. I'm not sure how to do this in a general form. By the way, I used to read about another way to prove this, namely by making representation with tensors (symmetrizing, antisymmetrizing and traceless conditions, like that). I wonder what's the connection between these 2 methods?

Answer 1

The direct product of spin-s representations can be decomposed into a direct sum of other spin-s representations using the highest weight decomposition. This decomposition can also be achieved by symmetrizing, anti symmetrizing, and imposing traceless conditions on tensors.

Step-by-step explanation:

The direct product of spin-s representations can be decomposed into a direct sum of other spin-s representations. This decomposition can be proved using the highest weight decomposition. The direct product of spin-s representations can be decomposed into a direct sum of other spin-s representations using the highest weight decomposition.

This decomposition can also be achieved by symmetrizing, antisymmetrizing, and imposing traceless conditions on tensors. The connection between this method and the method using tensors is that the decomposition into spin-s representations can also be achieved by symmetrizing, antisymmetrizing, and imposing traceless conditions on tensors.