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

Durfee squares, symmetric partitions and bounds on Kronecker coefficients