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 $\tilde{\mathcal{B}}(m)$ by adding a single generator to the usual Brauer category $\mathcal{B}(m)$, together with four relations. We prove that our category $\tilde{\mathcal{B}}(m)$ 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 $m$).