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%% feshbach.txt : summary of contents of feshbach.ps
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Let $\rho : G \hookrightarrow \GL(n, \F)$ be a representation of a finite group $G$
over the field $\F$\/, and let $\F[V]$ be the algebra of polynomial
functions on the vector space $V = \F^n$\/. The group $G$ acts on $V$ and
hence also on $\F[V]$ via the representation $\rho$ and we denote by
$\F[V]^G$ the subalgebra of $G$-invariant polynimials in $\F[V]$\/.
(We refer to \cite{poly} for basic facts about the invariant theory of
finite groups.) The {\bf transfer homomorphism}
\[
\Tr^G : \F[V] \longrightarrow \F[V]^G
\]
is defined by
\[
\Tr^G(f) = \sum_{g \in G} gf\comma\qquad \forall f \in \F[V]\period
\]
If the characteristic of $\F$ is relatively prime to the order of $G$ then
the transfer map is surjective. By contrast, if the characteristic of $\F$
divides the order of $G$ the image of the transfer, $\Im(\Tr^G)$\/, is an
ideal of height strictly less than $n$ \cite{Feshtwo}, \cite{kuhnigk}
\cite{survey}. We begin this note by reproving this result as well as M.
Feshbach's unpublished description of the radical of
$\Im(\Tr^G)$\/. We then go on to
describe the variety defined by the extended ideal $(\Im(\Tr^G))^e \subset
\F[V]$\/. It turns out that this variety has a particularly elegant
description, namely it is the union of the fixed point sets of the elements
of order $p$ in $G$\/, where $p$ is the characteristic of $\F$\/.
As an
application we show that the Dickson polynomial of least degree
$\dick_{n, n-1}$ is always a nonzero divisor in the quotient ring
$\F[V]/\Im(\Tr^G)$\/, so this quotient ring has positive homological
codimension (or depth).