Subject: Another question on invariant theory
From: "Douglas C. Ravenel" <doug@math.rochester.edu>
Date: Thu, 21 Jul 2005 11:39:12 -0400 (EDT)

Here is another question about invariant theory. 
Let the symmetric group $G=\Sigma_{p}$ act on the 
polynomial ring (over the integers) on $p$ variables 
in the usual way.  Then it acts on the set of monomials 
of degree $n$ via a permutation representation $\rho_{n}$.  
This information can be encoded in a Poincar\'e series 
with coefficients in the Burnside ring $A[G]$,

\begin{displaymath}
g (t) = \sum_{n \ge 0}\rho_{n}t^{n}.
\end{displaymath}

For $p=3$, $A[G]$ is generated by two elements, the 
regular representation $\rho $ of degree 3, and the 
standard representation $\sigma $ of degree 6.  
These are subject to the relations $\rho^{2}=\rho +\sigma$, 
$\rho \sigma =3\sigma $, and $\sigma^{2}=6\sigma$.

It is easy to compute $g (t)$ for $p=3$, and the answer is
\begin{displaymath}
1/g (t) = (1-t) (1-\alpha t+\beta t^{2}),
\end{displaymath}

\noindent where $\alpha =\rho -1$ and $\beta =\sigma -2\rho +1$.

My question is: Is there a similar formula for general $p$?  
Is there a conceptual (rather than computational) proof?

Doug



Douglas C. Ravenel

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