Breaking down the reduced Kronecker coefficients

Abstract

We resolve three interrelated problems on reduced Kronecker coefficients $\overline{g}(\alpha ,\beta ,\gamma )$. First, we disprove the 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 ${\textrm{\#P}}$-hard, i.e. ${\textrm{\#P}}$-hard when the input $(\lambda ,\mu ,\nu )$ is in unary.