The saturation conjecture for Littlewood–Richardson coefficients

Let $n$ be a positive integer, let $clambda$, $cmu$, and $cnu$ be partitions of length at most $n$, and let $T_n$ consist of the triples for which $V_\nu$ occurs in $V_\lambda\otimes V_\mu$, where $V_\lambda$ denotes the corresponding highest weight representation of $\operatorname{GL}_n(\mathbb C)$. For $N\in\mathbb N$, write $N\lambda$, $N\mu$, and $N\nu$ for the partitions obtained by scaling all parts by $N$. The saturation conjecture.

$$(\lambda,\mu,\nu)\in T_n\Longleftrightarrow(N\lambda,N\mu,N\nu)\in T_n.$$

This asserts that membership in the Littlewood–Richardson semigroup is unchanged by simultaneous dilation. It is equivalent to the saturation of the semigroup and was proved by Knutson and Tao using honeycombs, so the conjecture is solved.