Breaking down the reduced Kronecker coefficients

We resolve three interrelated problems on \emph{reduced Kronecker coefficients} $\overline{g}(\alpha,\beta,\gamma)$. First, we disprove the \emph{saturation property} which states that $\overline{g}(N\alpha,N\beta,N\gamma)>0$ implies $\overline{g}(\alpha,\beta,\gamma)>0$ for all $N>1$. Second, we esimate the maximal $\overline{g}(\alpha,\beta,\gamma)$, over all $|\alpha|+|\beta|+|\gamma| = n$. Finally, we show that computing $\overline{g}(\lambda,\mu,\nu)$ is strongly $\# P$-hard, i.e. $\#P$-hard when the input $(\lambda,\mu,\nu)$ is in unary.