= Tensor contractions in the SU2 Clebsch-Gordan decomposition
{title2=$V_m\otimes V_n=\bigoplus_{r=0}^{\min(m,n)}V_{m+n-2r}$}
Realize $V_n=\operatorname{Sym}^n\mathbb C^2$. Contract $r$ pairs of indices, one from each <symmetric tensor>, with the alternating <invariant tensor> $\epsilon$, and symmetrize the remaining indices. This is a nonzero <intertwining operator> into $V_{m+n-2r}$. In the bihomogeneous polynomial realization, $(z_1w_2-z_2w_1)^rz_1^{m-r}w_1^{n-r}$ is a <highest-weight vector> of weight $m+n-2r$. Its lowering string has length $m+n-2r+1$. These inequivalent irreducible summands exhaust the tensor product, since their dimensions sum to $(m+1)(n+1)$.
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