Subject: Question for AT list
From: "Douglas C. Ravenel" <doug@math.rochester.edu>
Date: Sat, 9 Dec 2006 16:34:06 -0500 (EST)

Dear Colleagues,

Here is a representation theory question which is not leiley to be new.  
Let $G$ be the cyclic group of order $p^{i+1}$ for $p$ a prime.  Write its 
group ring as

\begin{displaymath}
R=Z[G] = Z[x]/ (x^{p^{i+1}}-1)
\end{displaymath}

\noindent and consider the cyclic $R$-module

\begin{displaymath}
M=R/ (1+x^{p^{i}}+x^{2p^{i}}+\dotsb + x^{(p-1)p^{i}}).
\end{displaymath}

\noindent This module is isomorphic to the ring of integers in the 
$p^{i+1}$th cyclotomic field regarded as a module over the group of 
$p^{i+1}$th roots of unity.

I want to know the structure of the symmetric algebra $S (M)$ as a stable 
$R$-module, i.e. I am willing to ignore free $R$-summands in the 
description.  I already know the answer for $i=0$, and would be happy if 
someone could tell me the answer for $i=1$.

Doug

Douglas C. Ravenel

Department of Mathematics        |819 Hylan Building
University of Rochester            |(585) 275-4415
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Email:                 doug@math.removethis.rochester.&this.edu
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