Pham Huu Tiep

University of Florida
 
Title: Globally Irreducible Representations of Finite Groups and Integral
Lattices

Abstract:
  {The notion of globally irreducible representations of finite groups has been
introduced by B. H. Gross, in order to explain new series of Euclidean
lattices discovered recently by N. Elkies and T. Shioda using
Mordell-Weil lattices of elliptic curves. In this paper we first give a
necessary condition for global irreducibility. Then we classify all globally
irreducible representations of $L_{2}(q)$ and $^{2}B_{2}(q)$, and of majority 
of the $26$ sporadic finite simple groups. We also exhibit one more globally
irreducible representation, which is related to the Weil representation of
degree\break
$(p^{n}-1)/2$ of the symplectic group $Sp_{2n}(p)$
($p \equiv 1 (\bmod 4)$ is a prime). As a consequence, we get a new series of
even unimodular lattices of rank $2(p^{n}-1)$. A summary of currently known
globally irreducible representations is given.}

The paper has appeared in Geometriae Dedicata 64 (1997), 85 - 123.