MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Symmetric basis of harmonic homogeneus polynomials

Recently, a question about the beautiful theory of harmonic polynomials made me aware there is something I'd like to know for a long time.

As it is well known, for any number of variables $n$ and any degree $m$, there is a permutation invariant family of polynomials that generates linearly the space $\mathcal{H}_n^m$ of all (real) harmonic homogeneous polynomials in $n$ variables of degree $m$:

$$\mathcal{F}_ n^{\, m}:=\big\{ K\big(\partial^\alpha\|x\|^{2-n}\big)\ :\ \alpha\in\mathbb{N}^n\ ,\ \alpha_1+\dots+\alpha_n=m\ \big\}\ ,$$

where $K$ denotes the Kelvin transform, $Ku(x):=\|x\|^{2-n}u(\|x\|^{-2}x)$. Here, by permutation invariant set of polynomials I simply mean, if $P(x_1,\dots,x_n)$ is in it, then $P(x_{\sigma(1)},\dots,x_{\sigma(n)})$ also is in it, for any permutation $\sigma \in \mathfrak{S} _ n \ .$ The family $\mathcal{F}_ n^{\, m}$ is not linearly independent; however, taking only the elements corresponding to a suitable subset of multi-indices, namely, imposing e.g. $\alpha_n\le 1$, one actually finds a basis. What makes me a bit unhappy is, this constraint breaks the symmetry. So I wonder:

Is there an explicit permutation invariant subset of $\mathcal{F}_ n^{\, m}$ which is a basis of $\mathcal{H}_n^m$ ? If not always, for what $n$ and $m$? And, more generally, how to make an explicit permutation invariant basis ?

Actually, I do not see obstructions to the existence of such a basis, but I would also not be completely sure about how to prove its existence. Before starting and trying to answer by myself, I liked to pose the question here. My excuses if it turns out to be trivial!

-
If the result is false it can be disproved with a finite, albeit tedious, calculation. A finite-dimensional real vector space on which $S_n$ acts admits a permutation-invariant basis if and only if it is a direct sum of permutation representations, and for fixed $n$ this can be checked by comparing the character of the representation against the characters of all permutation representations (this is the tedious part). In particular the number of copies of the trivial representation is the number of orbits of the action of $S_n$, so if there are no such copies then the result is false. – Qiaochu Yuan Oct 20 2011 at 19:45
Thank you... So, the existence of an invariant basis may really depend on the pair $(n,m)$ ? – Pietro Majer Oct 20 2011 at 19:55
Sorry if I am being silly, but I don't see how the result you cite can be true for $n = 2$. – Qiaochu Yuan Oct 20 2011 at 22:25

Already the desired result is false for $n = 3, m = 2$, but for simpler reasons than I suggested in the comments. In this case the polynomials you give are $x^2 - 2y^2 + z^2, xy$ and their permutations. The sum of the permutations of $x^2 - 2y^2 + z^2$ is zero, so none of its permutations can be part of a permutation-invariant basis, and the span of the permutations of $xy$ don't span the entire space.