If we denote by $V(n-1)$ the n dimensional irreducible $\mathfrak{sl}_2$ module, thanks to the Clebsch Gordan rules, we can understand $V(n)\otimes V(m)$ as a direct sum of irreducible submodules.
This can be proved using the fact that in this situation character theory uniquely determines the representation theory. I.e 2 modules with the same character are isomorphic.
I encountered in James Humphrey's Lie algebra book the exercise to find a decomposition of $V(n)\otimes V(m)$ into a direct sum of irreducibles (chapter 7 exercise 6). My first reflex was, obviously, that this just meant proving the Clebsch Gordan rules. However I noticed that character theory had not at all been developped at this point.
So my question is, is there a way to prove the Clebsch Gordan rules, without any use of character theory?
Thanks in advance
Best Answer
Irreducible representations of $\mathfrak{sl}(2)$ are Lie algebra homomorphisms $\mathfrak{sl}(2)\to\mathfrak{gl}(V)$, and can be detected by where $h= \begin{pmatrix}1&\\&-1\end{pmatrix}\in\mathfrak{sl}(2)$ is sent. You can check that in the Lie algebra representation $\phi_n\colon\mathfrak{sl}(2)\to\mathfrak{gl}(V(n))$, the element $h$ is sent to the diagonal matrix $\mathrm{diag}(n,n-2,\dots,-n)$. In general, given a Lie algebra representation $\phi\colon\mathfrak{sl}(2)\to\mathfrak{gl}(V)$, the matrix $\phi(h)$ decomposes uniquely in terms of matrices of the form $\mathrm{diag}(n,n-2,\dots,-n)$, and this gives the decomposition of the representation $V$ in terms of the representations $V(n)$.
Now, to figure out how $V(m)\otimes V(n)$ decomposes, it suffices to see where $h$ is sent under the map $\phi_m\otimes\phi_n\colon \mathfrak{sl}(2)\to\mathfrak{gl}(V(m)\otimes V(n))$. But this is just $\mathrm{diag}(m,m-2,\dots,-m)\otimes 1+1\otimes\mathrm{diag}(n,n-2,\dots,-n)$, which, explicitly, has diagonal entries of the form $i+j$, where $m\ge i\ge -m$ and $n\ge j\ge -n$, where $i$ and $j$ have the same parities as $m$ and $n$, respectively. Now, you just have to determine how this decomposes in terms of the building blocks $\mathrm{diag}(\ell,\ell-2,\dots,-\ell)$.