Asymptotic and catalytic containment of representations of $\mathsf{SU}(n)$

Understand

Given two finite-dimensional representations $\rho$ and $\sigma$ of $\mathsf{SU}(n)$, when is there $n \in \mathbb{N}$ such that $\rho^{\otimes n}$ is isomorphic to a subrepresentation of $\sigma^{\otimes n}$? When is there a third representation $\eta$ such that $\rho \otimes \eta$ is a subrepresentation of $\sigma \otimes \eta$? We call these the questions of asymptotic and catalytic containment, respectively.

• We answer both questions in terms of an explicit family of inequalities.
• These inequalities are almost necessary and sufficient in the following sense.
• If two representations satisfy all inequalities strictly, then asymptotic and catalytic containment follow (the former in generic cases).