Let F be a field, and let Σn
be the symmetric group on n letters. In this paper
we address the following question: given two irreducible
FΣn-modules D1 and D2 of
dimensions greater than 1, can it happen that D1 [otimes ] D2
is irreducible? The answer is known to be ‘no’ if char F = 0
 (see also  for some generalizations). So we
assume from now on that F has positive characteristic p. The following conjecture
was made in .
CONJECTURE. Let D1 and D2 be
two irreducible FΣn-modules of dimensions
greater than 1. Then D1 [otimes ] D2
is irreducible if and only if p = 2, n = 2 + 4l for
some positive integer l; one of the modules corresponds to the partition
(2l + 2, 2l) and the other corresponds to a partition of the form
(n − 2j − 1, 2j + 1), 0 [les ] j < l.
Moreover, in the exceptional cases, one has
The main result of this paper is the following theorem, which establishes a big part
of the conjecture.