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\title{The minimal polynomials of unipotent elements in irreducible 
representations of the classical groups in characteristic $p$ with 
$p$-large highest weights}
\author{Irina D. Suprunenko \\
%EndAName
Institute of Mathematics\\
Academy of Sciences of Belarus\\
Minsk, 220072, Belarus\\
\date{}

\maketitle 

\centerline{{\bf Abstract}}

\bigskip
Irreducible representations of the classical groups over an 
algebraically closed field of odd characteristic $p$ are considered. 
The main goal of the paper is to show that the degree of the minimal 
polynomial of the image of a unipotent element of 
a classical group $G$ in an irreducible representation $\phi$
is equal to the order of this element provided the highest weight of $\phi$
is large enough with respect to $p$. More exactly, a function $l$ is 
defined on the set of irreducible representations of $G$; it is proved that the degree of the 
minimal polynomial of a unipotent element is equal to its order if 
$l(\phi)\ge p$, and 
for all four types of classical groups, arbitrarily large ranks and every 
$p$ examples of irreducible representations $\phi$ with $l(\phi)<p$ where the 
regular unipotent elements have the minimal polynomials of degrees less than 
their orders are given. For the special linear 
group the relevant results follow from the algorithms for computing the 
minimal polynomials of unipotent elements in irreducible representations 
announced by the author earlier. If $\phi$ is $p$-restricted, then 
$l(\phi)$ is equal to the value of its highest weight on the maximal 
root of the group. Some other results on the minimal polynomials of unipotent 
elements in irreducible representations of 
the symplectic group are obtained.

\bigskip
This is Preprint No~6 (1997) of the Institut fur Experimentelle Mathematik,
Univ. Essen
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