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Definition. Let $\text{Harm}(\mathbb{R}^n, G)$ be the space of $G$-harmonic polynomials on $\mathbb{R}^n$.

My question is, what is the dimension of $\text{Harm}(\mathbb{R}^n, S_n)$?

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  • $\begingroup$ It is equal to $n!$. See, for example, Talk 6 in Math Talks for Undergraduates by Serge Lang. $\endgroup$
    – Vladimir Dotsenko
    Dec 8, 2015 at 12:31

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