We investigate the question
when the tensor square, the alternating square, or the symmetric square of an
absolutely irreducible projective representation V of an almost simple group G is
again irreducible. The knowledge of such representations is of importance in the
description of the maximal subgroups of simple classical groups of Lie type. We show
that if G is of Lie type in odd characteristic, either V is a Weil representation of a
symplectic or unitary group, or G is one of a finite number of exceptions. For G in
even characteristic, we derive upper bounds for the dimension of V which are close to
the minimal possible dimension of nontrivial irreducible representations.
Our results are complete in the case of complex representations. We will
also answer a question of B. H. Gross about finite subgroups of complex
Lie groups G that act irreducibly on all fundamental representations of
G.
