The spin Brauer category (Q6652248)

One of the most classical results in representation theory is the Schur-Weyl duality, one half of which is the statement that the algebra homomorphism \(\mathbb{C}S_r\longrightarrow \text{End}_{\text{GL}(V)}(V^{\otimes r})\) is surjective, where \(S_r\) is the symmetric group on \(r\) letters acting on \(V^{\otimes r}\) by permutation of the factors. A more modern approach to the above involves considering morphisms between different powers of the natural module \(V\) to rephrase the results in terms of monoidal categories. P. J. McNamara and A. Savage introduce a diagrammatic monoidal category, the spin Brauer category, that plays the same role for the spin and pin groups as the Brauer category does for the orthogonal groups. That is, there is a full functor from the spin Brauer category to the category of finite-dimensional modules for the spin and pin groups. This functor becomes essentially surjective after passing to the Karoubi envelope, and its kernel is the tensor ideal of negligible morphisms. In this way, the spin Brauer category can be thought of as an interpolating category for the spin and pin groups. The authors also define an affine version of the spin Brauer category, which acts on categories of modules for the pin and spin groups via translation functors.