Knutson–Tao saturation conjecture

Let $\widehat{G}$ be a connected semisimple complex group, let $\widehat{T}$ be a maximal torus of $\widehat{G}$, and let $\mu_1,\mu_2,\lambda$ be weights of $\widehat{T}$. Suppose that $\mu_1+\mu_2+\lambda$ annihilates every element $s\in\widehat{T}$ whose centralizer in $\widehat{G}$ is a semisimple group. For a dominant weight $\mu$, write $V_\mu$ for the corresponding irreducible $\widehat{G}$-module. Knutson–Tao saturation conjecture. For every positive integer $N$,

$$(V_{N\mu_1}\otimes V_{N\mu_2}\otimes V_{N\lambda})^{\widehat{G}}\neq0\Rightarrow(V_{\mu_1}\otimes V_{\mu_2}\otimes V_\lambda)^{\widehat{G}}\neq0.$$

This conjecture proposes a sufficient condition for saturation of tensor-product invariants for general semisimple groups. The paper compares it with the stronger minuscule-weight saturation conjecture above; neither the comparison nor the conjecture is resolved in the supplied text.