I'd like to understand some special things about representations of $sl_2$ (which is considered as a Lie algebra over $\mathbb{C}$). First, it can be shown that for each $n\in \mathbb{N}$ there is only one $n$-dimensional irreducible representation of $sl_2$: let me denote it by $V_n$. Could you please help me to understand the following identities? $$\mathrm{Sym}^2 V_n=\bigoplus_{i=0}^{[n/2]} V_{2n-4i}$$ $$\mathrm{Sym}^k V_n\simeq \mathrm{Sym}^n V_k$$ I will be very grateful for either a hint or a link for a proof!
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I am not sure your identities are correct if $n$ is the dimension of the irreducible representation. Rather, I think $n$ should be the highest weight of the representation (for which the dimension is $n + 1$.) Taking a look at your second example, if the subscripts refer to dimensions of the vector spaces: $$\text{highest weight of $S^{k}V_{n} = k(n-1)$} \not = \text{$n(k-1) =$ highest weight of $S^{n}V_{k}$ }.$$ On the other hand, if the subscripts refer to the weights, considering the set of weights on both sides of the equation and trying to show that these sets are equal should be useful. – Siddharth Venkatesh Apr 24 '14 at 09:47
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In fact, for the second equality, the weights are really easy to calculate because it is easy to find a weight basis for the symmetric power based on the weight basis of the original representation. I can provide more details if you like but it's a really fun calculation. – Siddharth Venkatesh Apr 24 '14 at 09:48