  In this paper we study irreducible tensor products of representations of
alternating groups in characteristics 2 and 3. In characteristic 3 we
completely classify irreducible tensor products, while in characteristic 2 we
completely classify irreducible tensor products where none of the modules
appearing in the product is a basic spin module. In characteristic 2 we also
give some necessary conditions for the tensor product of an irreducible module
with a basic spin module to be irreducible.
