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Let $G$ be a finite group acting on a vector space $V$. Let $\mathcal{D}$ denote the differential operators with constant coefficients and $\mathcal{D}^{G}$ be the $G$-invariant operators. A polynomial $p$ is called harmonic if $D(p)=0$ for all $D\in\mathcal{D}^G$. Denote the set of harmonic polynomials by $H_G$. In the case of the symmetric group these polynomials are spanned by the Vandermonde determinant and all its partial derivatives. This result seems to generalize to all finite reflection groups. For a general finite group, it is clear that $H_G$ is a finite dimensional vector space, but is there a similar construction to obtain a basis?

Let $G$ be a finite group acting on a complex vector space $V$. Let $\mathcal{D}$ denote the differential operators with constant coefficients and $\mathcal{D}^{G}$ be the $G$-invariant operators. A polynomial $p$ is called harmonic if $D(p)=0$ for all $D\in\mathcal{D}^G$. Denote the set of harmonic polynomials by $H_G$. In the case of the symmetric group these polynomials are spanned by the Vandermonde determinant and all its partial derivatives. This result seems to generalize to all finite reflection groups. For a general finite group, it is clear that $H_G$ is a finite dimensional vector space, but is there a similar construction to obtain a basis?

Let $G$ be a finite group and $H_G$ be the set of harmonic polynomials. In the case of the symmetric group these polynomials are spanned by the Vandermonde determinant and all its partial derivatives. This result seems to generalize to all finite reflection groups. For a general finite group, it is clear that $H_G$ is a finite dimensional vector space, but is there a similar construction to obtain a basis?

Let $G$ be a finite group acting on a vector space $V$. Let $\mathcal{D}$ denote the differential operators with constant coefficients and $\mathcal{D}^{G}$ be the $G$-invariant operators. A polynomial $p$ is called harmonic if $D(p)=0$ for all $D\in\mathcal{D}^G$. Denote the set of harmonic polynomials by $H_G$. In the case of the symmetric group these polynomials are spanned by the Vandermonde determinant and all its partial derivatives. This result seems to generalize to all finite reflection groups. For a general finite group, it is clear that $H_G$ is a finite dimensional vector space, but is there a similar construction to obtain a basis?

Let $G$ be a finite group and $H_G$ be the set of harmonic polynomials. In the case of the symmetric group these polynomials are spanned by the Vandermonde determinant and all its partial derivatives. This result seems to generalize to all finite reflection groups. However, is there a similar concept for other finite groups?

Let $G$ be a finite group and $H_G$ be the set of harmonic polynomials. In the case of the symmetric group these polynomials are spanned by the Vandermonde determinant and all its partial derivatives. This result seems to generalize to all finite reflection groups. For a general finite group, it is clear that $H_G$ is a finite dimensional vector space, but is there a similar construction to obtain a basis?