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 $\mathrm{SO}(2n+1)$, $\mathrm{Sp}(2n)$ and $\mathrm{SO}(2n)$. 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 $\mathbb{S}_{\langle\lambda\rangle}(G)$ for $G$ with type $C_n$. Here $\mathbb{S}_{[\lambda]}$ and $\mathbb{S}_{\langle\lambda\rangle}$ are the usual Schur functors associated to the partition $\lambda$.