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An orthonormal basis for the space of harmonic polynomials in $n$ variables is provided by the spherical harmonics on the $n-1$ sphere, see e.g. wiki.

From there, constructing an orthonormal basis for the subspace of harmonic polynomials symmetric in $n$ variables is straightforward but laborious (or is it?). Are there known closed expressions, or simple derivation, for such a basis?

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    $\begingroup$ Related mathoverflow.net/questions/78694/… ? $\endgroup$
    – Nemo
    Commented Apr 4 at 15:09
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    $\begingroup$ With $n=2$ it seems reasonably simple to derive a basis for the harmonic symmetric polynomials with one polynomial of each degree except 2, and then normalise them with Gram-Schmidt. With $n=3$ it becomes more complicated because there are multiple independent basis polynomials of some degrees. $\endgroup$
    – Peter Taylor
    Commented Apr 5 at 8:42

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