Kronecker multiplicities in the $(k,\ell)$ hook are polynomially bounded

The problem of decomposing the Kronecker product of $S_n$ characters is one of the last major open problems in the ordinary representation theory of the symmetric group $S_n$. Here we prove upper and lower polynomial bounds for the multiplicities of the Kronecker product $χ^\lm\otimesχ^μ$, where for some fixed $k$ and $\ell$ both partitions $\lm$ and $μ$ are in the $(k,\ell)$ hook, $\lm$ and $μ$ are partitions of $n$, and $n$ goes to infinity.

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