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?
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What tools do you have at your disposal? – Cheerful Parsnip Jun 16 '14 at 19:25
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Hm... I suppose only weights. Also I imagine what a character is. – Kolyan Jun 16 '14 at 19:38
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For example it follows from the Cayley-Sylvester formula (http://www.sciencedirect.com/science/article/pii/S0195669806000916) that actually tells you the decomposition of $S^n(S^mV)$ into irreducible representation. Probably there is a more direct way but I don't know it. – Daniele A Jun 16 '14 at 21:07