Invariants of the special orthogonal group and an enhanced Brauer category

TL;DR

This paper provides a diagrammatic proof of the First Fundamental Theorem for the special orthogonal group, introduces an enhanced Brauer category, and proves its equivalence to the category of SO_m representations, enabling new computational methods.

Contribution

It introduces an enhanced Brauer category with generators and relations, proving its equivalence to the SO_m representation category, and offers a diagrammatic approach to invariant theory.

Findings

Proved the FFT for SO_m using a diagrammatic method.

Defined an enhanced Brauer category equivalent to SO_m representations.

Established a framework for computing homomorphism dimensions in tensor modules.

Abstract

We first give a short intrinsic, diagrammatic proof of the First Fundamental Theorem of invariant theory (FFT) for the special orthogonal group $\text{SO}_m(\mathbb{C})$, given the FFT for $\text{O}_m(\mathbb{C})$. 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) {\em equivalent} to the category of representations of $\text{SO}_m$ generated by the natural representation. The FFT for $\text{SO}_m$ amounts to the surjectivity of a certain functor $\mathcal{F}$ on $\text{Hom}$ spaces, while the Second Fundamental Theorem for $\text{SO}_m$ says simply that $\mathcal{F}$ is injective on $\text{Hom}$ spaces. This theorem provides a diagrammatic means of computing the dimensions of spaces of homomorphisms between tensor modules for $\text{SO}_m$ (for any $m$). These methods will be applied to the case of the orthosymplectic Lie algebras $\text{osp}(m|2n)$, where the super-Pfaffian enters, in a future work.