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 ).