Irreducible representation of tensor product

Question

Let $\varphi_n$ denote the standart irreducible representation of $SU(2)$ group with highest weight $n$.

What are the irreducible representaions of $\varphi_2 \otimes \varphi_3$?

Answer 1

A result commonly attributed to Clebsch and Gordan gives you the answer: $$ \varphi_2\otimes \varphi_3\cong \varphi_5\oplus\varphi_3\oplus\varphi_1. $$ You can study the dimensions of weight spaces to see this. The weight five space is 1-dimensional (spanned by the tensor product of the highest weight weight spaces of both $\varphi_2$ and $\varphi_3$. The weight three space is two dimensional (= direct sum of two tensor products. That of weight zero space of $\varphi_2$ and weight three space of $\varphi_3$, and that of weight two space of $\varphi_2$ and weight one space of $\varphi_3$). The summand generated by the weight five submodule consumes only one of the two, leaving us with a high weight vector of weight three unaccounted for - resulting in a summand $\varphi_3$ on the r.h.s.. Continuing in the same vein leaves us with an extra $\varphi_1$ component.

Let's check the dimensions. $\dim\varphi_n=n+1$, so the tensor product $\varphi_2\otimes\varphi_3$ is of dimension $3\cdot4=12$. On the r.h.s we have summands of dimensions $6+4+2=12$. Check.

In quantum mechanics this result is known as the quantum mechanical addition of angular momenta.

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The general formula for the tensor product of two irreducible reps of $SU_2$ (or $sl_2$ reads: (assume $m\ge n$) $$ \varphi_m\otimes\varphi_n=\varphi_{m+n}\oplus\varphi_{m+n-2}\oplus\varphi_{m+n-4}\oplus\cdots\oplus\varphi_{m-n+2}\oplus\varphi_{m-n} $$