# Schwarz (1975) theorem

theorem:

Let $\Gamma$ be a compact Lie group acting on $V$. Let $u_{1},\ldots,u_{s}$ be a Hilbert basis for the $\Gamma$-invariant polynomials $\mathcal{P}(\Gamma)$ (see Hilbert-Weyl theorem). Let $f\in\mathcal{E}(\Gamma)$. Then there exists a smooth germ $h\in\mathcal{E}_{s}$ (the ring of $C^{\infty}$ germs $\mathbb{R}^{s}\to\mathbb{R}$) such that $f(x)=h(u_{1}(x),\ldots,u_{s}(x))$. [GSS]

proof:

The proof is shown on page 58 of [GSS].

theorem: (as stated by Gerald W. Schwarz)

Let $G$ be a compact Lie group acting orthogonally on $\mathbb{R}^{n}$, let $\rho_{1},\ldots,\rho_{k}$ be generators of $\mathcal{P}(\mathbb{R}^{n})^{G}$ (the set $G$-invariant polynomials on $\mathbb{R}^{n}$), and let $\rho=(\rho_{1},\ldots,\rho_{k}):\mathbb{R}^{n}\to\mathbb{R}^{k}$. Then $\rho*\mathcal{E}(\mathbb{R}^{k})=\mathcal{E}(\mathbb{R}^{n})^{G}$. [SG]

proof:

The proof is shown in the following publication [SG].

## References

• GSS Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David: Singularities and Groups in Bifurcation Theory (Volume II). Springer-Verlag, New York, 1988.
• SG Schwarz, W. Gerald: Smooth Functions Invariant Under the Action of a Compact Lie Group, Topology Vol. 14, pp. 63-68, 1975.
Title Schwarz (1975) theorem Schwarz1975Theorem 2013-03-22 13:40:06 2013-03-22 13:40:06 mathcam (2727) mathcam (2727) 9 mathcam (2727) Theorem msc 13A50