On Classical groups detected by the tensor third representation

Motivated by the Langlands’ beyond endoscopy proposal for establishing functoriality, we study the representation ${\otimes }^3$ in a setting related to the Langlands $L$-functions $L(s,\pi ,{\otimes }^3),$ where $\pi$ is a cuspidal automorphic representation of $G$ where $G$ is either $\mathrm{S}\mathrm{O}(2n+1)$, $\mathrm{S}\mathrm{p}(2n)$ and $\mathrm{S}\mathrm{O}(2n)$. In particular, under what conditions on partitions $\lambda$, we examine whether or not ${\otimes }^3$ detects the subgroups ${\mathbb{S}}_{[\lambda ]}(G)$ for $G$ with type $B_n$ and $D_{2n}$ or ${\mathbb{S}}_{⟨\lambda ⟩}(G)$ for $G$ with type $C_n$. Here ${\mathbb{S}}_{[\lambda ]}$ and ${\mathbb{S}}_{⟨\lambda ⟩}$ are the usual Schur functors associated to the partition $\lambda$.