Durfee squares, symmetric partitions and bounds on Kronecker coefficients

Abstract

We resolve two open problems on Kronecker coefficients g(λ,μ,ν) of the symmetric group. First, we prove that for partitions λ,μ,ν with fixed Durfee square size, the Kronecker coefficients grow at most polynomially. Second, we show that the maximal Kronecker coefficients g(λ,λ,λ) for self-conjugate partitions λ grow superexponentially. We also give applications to explicit special cases.