A new generalization of Hermite's reciprocity law

Abstract

Given a partition $\lambda$ of $n$, the {\it Schur functor} ${\mathbb{S}}_{\lambda }$ associates to any complex vector space $V$, a subspace ${\mathbb{S}}_{\lambda }(V)$ 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 ${\mathbb{S}}_{\lambda }\left({\mathbb{S}}_{\delta }({\mathbb{C}}^2)\right)\simeq {\mathbb{S}}_{\mu }\left({\mathbb{S}}_{\epsilon }({\mathbb{C}}^2)\right)$.

arXiv commentAccepted in Journal of Algebraic Combinatorics