# Hilbert-Weyl theorem

Theorem: Let $\Gamma$ be a compact Lie group acting on $V$. Then there exists a finite Hilbert basis for the ring $\mathcal{P}(\Gamma)$ (the set of invariant polynomials). [GSS]

proof:

In [GSS] on page 54.

Theorem:(as stated by Hermann Weyl)

The (absolute) invariants corresponding to a given set of representations of a finite or a compact Lie group have a finite integrity basis. [HW]

proof:

In [HW] on page 274.

## References

• GSS Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David.: Singularities and Groups in Bifurcation Theory (Volume II). Springer-Verlag, New York, 1988.
• HW Hermann, Weyl: The Classical Groups: Their Invariants and Representations. Princeton University Press, New Jersey, 1946.
Title Hilbert-Weyl theorem HilbertWeylTheorem 2013-03-22 13:39:54 2013-03-22 13:39:54 mathcam (2727) mathcam (2727) 9 mathcam (2727) Theorem msc 22E20