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

TL;DR

This paper characterizes when one finite-dimensional representation of SU(n) can be asymptotically or catalytically contained within another using explicit inequalities, advancing understanding of representation containment.

Contribution

It provides explicit inequalities that nearly fully characterize asymptotic and catalytic containment of SU(n) representations, connecting to the Vergleichsstellensatz.

Findings

Inequalities nearly characterize asymptotic containment

Inequalities nearly characterize catalytic containment

Results connect to the Vergleichsstellensatz

Abstract

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). Conversely, if asymptotic or catalytic containment holds, then the inequalities must hold non-strictly. These results are an instance of a recent \emph{Vergleichsstellensatz} applied to the representation semiring.