Schwarz (1975) theorem


Let Γ be a compact Lie group acting on V. Let u1,,us be a Hilbert basis for the Γ-invariant polynomials 𝒫(Γ) (see Hilbert-Weyl theorem). Let f(Γ). Then there exists a smooth germ hs (the ring of C germs RsR) such that f(x)=h(u1(x),,us(x)). [GSS]


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 n, let ρ1,,ρk be generatorsPlanetmathPlanetmathPlanetmath of 𝒫(n)G (the set G-invariant polynomials on Rn), and let ρ=(ρ1,,ρk):nk. Then ρ*(k)=(n)G. [SG]


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


  • 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
Canonical name Schwarz1975Theorem
Date of creation 2013-03-22 13:40:06
Last modified on 2013-03-22 13:40:06
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Theorem
Classification msc 13A50