Hermit reciprocity, $\mathfrak{sl}_2(\mathbb{C})$

Question

Let $V$ be the standard $2$-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$. Hermit reciprocity states that $S^n(S^mV)\simeq S^m(S^nV)$. Can anybody give me a hint to prove it or give a reference?