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

As 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 a 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 that 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'd like to pose the question here. My apologies if it turns out to be trivial!

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

As 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 a 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 that 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'd like to pose the question here. My apologies if it turns out to be trivial!

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

As 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 a 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 that 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'd like to pose the question here. My apologies if it turns out to be trivial!

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

As 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 a 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 that 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'd like to pose the question here. My apologies if it turns out to be trivial!

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!

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

As 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 a 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 that 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'd like to pose the question here. My apologies if it turns out to be trivial!