Durfee squares, symmetric partitions and bounds on Kronecker coefficients

Abstract

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.

arXiv comment16 pages, 1 figure. Version 2 adds details on some calculations, fixed typos, improved exposition