A new generalization of Hermite’s reciprocity law

Given a partition $$\lambda $$? of n, the Schur functor$${\mathbb {S}}_\lambda $$S? associates to any complex vector space V, a subspace $${\mathbb {S}}_\lambda (V)$$S?(V) of $$V^{\otimes n}$$V?n. Hermite's reciprocity law, in terms of the Schur functor, states that $${\mathbb {S}}_{(p)}\left( {\mathbb {S}}_{(q)}({\mathbb {C}}^2)\right) \simeq {\mathbb {S}}_{(q)}\left( {\mathbb {S}}_{(p)}({\mathbb {C}}^2)\right) . $$S(p)S(q)(C2)?S(q)S(p)(C2). We extend this identity to many other identities of the type $${\mathbb {S}}_{\lambda }\left( {\mathbb {S}}_{\delta }({\mathbb {C}}^2)\right) \simeq {\mathbb {S}}_{\mu }\left( {\mathbb {S}}_{\epsilon }({\mathbb {C}}^2)\right) $$S?S?(C2)?SμS∈(C2).