On a theorem of Barbara Schmid

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 $G$.