Invariants of the special orthogonal group and an enhanced Brauer category

  • Gustav I. Lehrer

    University of Sydney, Australia
  • Ruibin Zhang

    University of Sydney, Australia
Invariants of the special orthogonal group and an enhanced Brauer category cover
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Abstract

We first give a short intrinsic, diagrammatic proof of the First Fundamental Theorem of invariant theory (FFT) for the special orthogonal group SOm(C)_m(\mathbb C), given the FFT for Om(C)_m(\mathbb C). We then define, by means of a presentation with generators and relations, an enhanced Brauer category B~(m)\widetilde{\mathcal B}(m) by adding a single generator to the usual Brauer category B(m)\mathcal B(m), together with four relations. We prove that our category B~(m)\widetilde{\mathcal B}(m) is actually (and remarkably) equivalent to the category of representations of SOm_m generated by the natural representation. The FFT for SOm_m amounts to the surjectivity of a certain functor F\mathcal F on Hom spaces, while the Second Fundamental Theorem for SOm_m says simply that F\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 SOm_m (for any mm).

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. 181–200

DOI 10.4171/LEM/63-1/2-6