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 ${\text{SO}}_m(\mathbb{C})$, 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 $\tilde{\mathcal{B}}(m)$ 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) {\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.

References in corpus (2)

• Invariants of the orthosymplectic Lie superalgebra and super Pfaffians
• The second fundamental theorem of invariant theory for the orthogonal group