Subject: Question for AT list
From: "Douglas C. Ravenel"
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
Rochester, New York 14627 |FAX (585) 273-4655
Email: doug@math.removethis.rochester.&this.edu
Personal home page: http://www.math.rochester.edu/people/faculty/doug/
Department home page: http://www.math.rochester.edu/
Math 237 home page: http://www.math.rochester.edu/courses/237/home/