Breaking down the reduced Kronecker coefficients

We resolve three interrelated problems on \emph{reduced Kronecker coefficients} $\overline{g}(α,β,γ)$. First, we disprove the \emph{saturation property} which states that $\overline{g}(Nα,Nβ,Nγ)>0$ implies $\overline{g}(α,β,γ)>0$ for all $N>1$. Second, we esimate the maximal $\overline{g}(α,β,γ)$, over all $|α|+|β|+|γ| = n$. Finally, we show that computing $\overline{g}(λ,μ,ν)$ is strongly $\# P$-hard, i.e. $\#P$-hard when the input $(λ,μ,ν)$ is in unary.

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