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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.
In this paper we present a categorical version of the first and second fundamental theorems of the invariant theory for the quantized symplectic groups. Our methods depend on the theory of braided strict monoidal categories which are pivotal, more explicitly the diagram category of framed tangles.
We define an infinite chain of subcategories of the partition category by introducing the left-height ($l$) of a partition. For the Brauer case, the chain starts with the Temperley-Lieb ($l=-1$) and ends with the Brauer ($l=infty$) category. The End
Let $G$ be a real classical group of type $B$, $C$, $D$ (including the real metaplectic group). We consider a nilpotent adjoint orbit $check{mathcal O}$ of $check G$, the Langlands dual of $G$ (or the metaplectic dual of $G$ when $G$ is a real metapl
We prove the twisted Whittaker category on the affine flag variety and the category of representations of the mixed quantum group are equivalent.
Let C_n denote the representation category of a finite supergroup generated by purely odd n-dimensional vector space. We compute the Brauer-Picard group BrPic(C_n) of C_n. This is done by identifying BrPic(C_n) with the group of braided tensor autoeq