Invariants of the special orthogonal group and an enhanced Brauer category
Gustav I. LehrerUniversity of Sydney, Australia
Ruibin ZhangUniversity of Sydney, Australia
We first give a short intrinsic, diagrammatic proof of the First Fundamental Theorem of invariant theory (FFT) for the special orthogonal group SO, given the FFT for O. We then define, by means of a presentation with generators and relations, an enhanced Brauer category by adding a single generator to the usual Brauer category , together with four relations. We prove that our category is actually (and remarkably) equivalent to the category of representations of SO generated by the natural representation. The FFT for SO amounts to the surjectivity of a certain functor on Hom spaces, while the Second Fundamental Theorem for SO says simply that is injective on Hom spaces. This theorem provides a diagrammatic means of computing the dimensions of spaces of homomorphisms between tensor modules for SO (for any ).
Cite this article
Gustav I. Lehrer, Ruibin Zhang, Invariants of the special orthogonal group and an enhanced Brauer category. Enseign. Math. 63 (2017), no. 1/2, pp. 181–200DOI 10.4171/LEM/63-1/2-6