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.