Return to Revisions

It is classical that, as $O(n)$-representation, $$ \text{Sym}^k(\mathbf R^n)=H^k\oplus qH^{k-2}\oplus q^2H^{k-4}\oplus\ldots $$ Here $q=x_1^2+\ldots+x_n^2$ is the quadratic form defining $O(n)$ and $H^k\subseteq \text{Sym}^k(\mathbf R^n)$ is the space of harmonic polynomials, i.e., polynomials which are killed by the Laplacian $\Delta=\partial_{x_1}^2+\ldots+\partial_{x_n}^2$. Moreover all spaces $q^lH^k$ are irreducible.