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Let $V$ be the standard representation of $S_3$. Show that $\text{Sym}^{k+6}V\cong \text{Sym}^kV\oplus \mathbb{k} S_3$.

This is exercise 1.12 from Fulton and Harris. There's a solution here that follows the notation set up in Fulton and Harris with $\mathbb{k}=\mathbb{C}$. However, I'd like to find a solution for arbitrary field $\mathbb{k}$ (with characteristic not dividing $6$). Perhaps a character-theoretic proof?

I tried working out the characters but didn't get far. Any help is appreciated.

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