Langlands’ beyond endoscopy proposal for establishing
functoriality motivates the study of irreducible subgroups of
$\mathrm{GL}_n$ that stabilize a line in a given representation of
$\mathrm{GL}_n$. Such subgroups are said to be detected by the
representation. In this talk we present a family of results when the
subgroup is a classical group in the important special case where the
representation of $\mathrm{GL}_n$ is a subrepresentation of the triple
tensor product representation $\otimes^3$. If time permits, we will
prove some partition identities arising in this context and discuss
their relation to explicit Satake inversion.