Invariants of the special orthogonal group and an enhanced Brauer category

Abstract

We first give a short intrinsic, diagrammatic proof of the First Fundamental Theorem of invariant theory (FFT) for the special orthogonal group SO$_m(\mathbb C)$, given the FFT for O$_m(\mathbb C)$. We then define, by means of a presentation with generators and relations, an enhanced Brauer category $\widetilde{\mathcal B}(m)$ by adding a single generator to the usual Brauer category $\mathcal B(m)$, together with four relations. We prove that our category $\widetilde{\mathcal B}(m)$ is actually (and remarkably) equivalent to the category of representations of SO$_m$ generated by the natural representation. The FFT for SO$_m$ amounts to the surjectivity of a certain functor $\mathcal F$ on Hom spaces, while the Second Fundamental Theorem for SO$_m$ says simply that $\mathcal F$ is injective on Hom spaces. This theorem provides a diagrammatic means of computing the dimensions of spaces of homomorphisms between tensor modules for SO$_m$ (for any $m$).