Unimodality via Kronecker coefficients

Sylvester's classical theorem states that q-binomial coefficients are unimodal (as polynomials in q). Despite a number of algebraic and combinatorial proofs, this result remains largely mysterious. I will present several unimodality results extending Sylvester's theorem which do not seem attainable by any known techniques. Notably, we establish strict(!) unimodality of large enough q-binomial coefficients. The proofs are based on properties of the Kronecker coefficients of the symmetric group. Joint work with Greta Panova.