A new generalization of Hermite's reciprocity law

Given a partition $λ$ of $n$, the {\it Schur functor} $\mathbb{S}_λ$ associates to any complex vector space $V$, a subspace $\mathbb{S}_λ(V)$ of $V^{\otimes n}$. Hermite's reciprocity law, in terms of the Schur functor, states that $\mathbb{S}_{(p)}\left(\mathbb{S}_{(q)}(\mathbb{C}^2)\right)\simeq \mathbb{S}_{(q)}\left(\mathbb{S}_{(p)}(\mathbb{C}^2)\right).$ We extend this identity to many other identities of the type $\mathbb{S}_λ\left(\mathbb{S}_δ(\mathbb{C}^2)\right)\simeq \mathbb{S}_μ\left(\mathbb{S}_ε(\mathbb{C}^2)\right)$.

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