  We study the remarkable Saxl conjecture which states that tensor squares of
certain irreducible representations of the symmetric groups S_n contain all
irreducibles as their constituents. Our main result is that they contain
representations corresponding to hooks and two row Young diagrams. For that, we
develop a new sufficient condition for the positivity of Kronecker coefficients
in terms of characters, and use combinatorics of rim hook tableaux combined
with known results on unimodality of certain partition functions. We also
present connections and speculations on random characters of S_n.
