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Given a partition λ of n, the Schur functor S λ associates to any complex vector space V , a subspace S λ (V ) of V ⊗n . Hermite's reciprocity law, in terms of the Schur functor, states that S (p) S (q) (C 2 ) ≃ S (q) S (p) (C 2 ) . We extend this identity to many other identities of the type S λ S δ (C 2 ) ≃ Sµ Sǫ(C 2 ) .

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In this article, we formally define and investigate the computational complexity of the definability problem for open first-order formulas (i.e. quantifier free first-order formulas) with equality. Given a logic $\boldsymbol{\mathcal{L}}$, the $\boldsymbol{\mathcal{L}}$-definability problem for finite structures takes as an input a finite structure $\boldsymbol{A}$ and a target relation $T$ over the domain of $\boldsymbol{A}$ and determines whether there is a formula of $\boldsymbol{\mathcal{L}}$ whose interpretation in $\boldsymbol{A}$ coincides with $T$. We show that the complexity of this problem for open first-order formulas (open definability, for short) is coNP-complete. We also investigate the parametric complexity of the problem and prove that if the size and the arity of the target relation $T$ are taken as parameters, then open definability is $\textrm{coW}[1]$-complete for every vocabulary $\tau $ with at least one, at least binary, relation.

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