A new generalization of Hermite's reciprocity law

TL;DR

This paper extends Hermite's reciprocity law, a fundamental symmetry involving Schur functors, to a broader class of identities, revealing new algebraic equivalences in the representation theory of complex vector spaces.

Contribution

It generalizes Hermite's reciprocity law to a wider set of identities involving Schur functors and complex vector spaces.

Findings

Extended Hermite's reciprocity to new identities

Established algebraic equivalences between different Schur functor compositions

Provided a framework for further generalizations in representation theory

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 $\mathbb{S}_{\lambda}\left(\mathbb{S}_{\delta}(\mathbb{C}^2)\right)\simeq \mathbb{S}_{\mu}\left(\mathbb{S}_{\epsilon}(\mathbb{C}^2)\right)$.