Durfee squares, symmetric partitions and bounds on Kronecker coefficients

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