
\titleofpresentation{Hartley's theorem on representations of the
  general linear groups: a new approach} 

\presenter{A. E. Zalesski}  %  Name just one speaker, please; collaborators
			%  can be credited in the body of the abstract.

In 1986 Brian Hartley \cite{Zalesski} obtained the following
interesting result: 

\begin{theorem}  Let $K$ be a field, $E$ 
the standard $GL_n(K)$-module, and $V$ be an irreducible
finite-dimensional $GL_n(K)$-module over $K$ with $\dim V>1$.  
 Let $H$ be a finite subgroup of $ 
GL_n(K)$. Suppose that $H$ is not of exponent $2$ and the restriction
of $E$ to $H$ contains a free  
$KH$-module. Then $V$ contains a free $KH$-submodule. 
\end{theorem}

His proof is based on deep properties of the duality between
irreducible representations of the general linear group $GL_n(K)$ and
the symmetric group $S_n$. We suggest another proof based on general
principles of representation theory of algebraic groups, more
precisely, on properties of the weight systems of algebraic group
representations. Therefore, it seems to be more conceptual and in a
sense more transparent. We also expect that our method will have
further applications. 

%  [Body of abstract goes here.]

\begin{ourbiblio}{99}  %  Delete this environment if you
		       %  aren't giving a bibliography.
\bibitem{Zalesski} B. Hartley, Relative higher relations modules, and a
property of irreducibble $K$-representations of $GL_n(K)$, J.
Algebra 104(1986),  113--125.
%\bibitem{yourname1} One paper, keyed to your name to avoid conflicts 
%with other abstracts;

 
\end{ourbiblio}

\presenteraddress{University of East Anglia, Norwich, UK}  % The name of a university is fine.

\email{h054@uea.ac.uk}
  % Those without homepages should delete this.

% Delete the next line when you are done.


