\title{Some conjectures and their consequences for tensor products of modules over a finite $p$-group}

\author{Dave Benson}
\address{Institute of Mathematics, University of Aberdeen, Aberdeen
  AB24 3UE, Scotland, United Kingdom}

\begin{document}

\maketitle

\begin{abstract}
We make some conjectures about tensor products of 
modular representations for a finite $p$-group in
characteristic $p$, for $p=2$ and $p=3$. The easiest
to state is that if $M$ is a module of odd dimension
for a $2$-group over an algebraically closed field of 
characteristic two, then $k$ is the only direct summand
of $M\otimes M^*$ of odd dimension, and the remaining
indecomposable summands have dimension divisible
by four. As a consequence, the odd dimensional modules
form an abelian group, where the composition law is to
take the unique indecomposable summand of odd dimension
of the tensor product, and where the inverse of an odd
dimensional module is its dual. In characteristic three
the conjectures are more complicated, and in larger 
characteristics it is not at all clear what the corresponding
conjectures should be.
\end{abstract}