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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a theorem of Barbara Schmid


Author: Larry Smith
Journal: Proc. Amer. Math. Soc. 128 (2000), 2199-2201
MSC (1991): Primary 13A50
Published electronically: November 29, 1999
MathSciNet review: 1654096
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Abstract: Let $G$ be a finite group and $\rho\colon\ G\hookrightarrow\mathrm{GL} (n,\mathbb{C})$ a complex representation. Barbara Schmid has shown that the algebra of invariant polynomial functions $\mathbb{C}[V]^G$ on the vector space $V=\mathbb{C}^n$ is generated by homogeneous polynomials of degree at most $\beta$, where $\beta$ is the largest degree of a generator in a minimal generating set for $\mathbb{C}[\mathrm{reg}_{\mathbb{C}}(G)]^G$, and $\mathrm{reg}_{\mathbb{C}}(G)$ is the complex regular representation of $G$. In this note we give a new proof of this result, and at the same time extend it to fields $\mathbb{F}$ whose characteristic $p$ is larger than $|G|$, the order of the group $G$.


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Additional Information

Larry Smith
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455; Mathematisches Institut der Universität, D 37073 Göttingen, Germany
Email: smith@math.umn.edu, larry@sunrise.uni-math.gwdg.de

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05259-4
PII: S 0002-9939(99)05259-4
Published electronically: November 29, 1999
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2000 American Mathematical Society