Kronecker products, characters, partitions, and the tensor square conjectures

We study the remarkable Saxl conjecture which states that tensor squares of certain irreducible representations of the symmetric groups Sn contain all irreducibles as their constituents. Our main result is that for sufficiently large n they contain representations corresponding to Young diagrams of hooks, two row and diagrams obtained from hooks and two rows by adding a finite number of squares. For that, we develop a new sufficient condition for the positivity of Kronecker coefficients in terms of characters, and apply this tool using combinatorics of rim hook tableaux in combination with known results on unimodality of certain partition functions. We also present connections and speculations on random characters of Sn.