Abstract
We define the affine VW supercategory 
, which arises from studying the action of the periplectic Lie superalgebra \(\mathfrak {p}(n)\) on the tensor product \(M\otimes V^{\otimes a}\) of an arbitrary representation M with several copies of the vector representation V of \(\mathfrak {p}(n)\). It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group; the main obstacle was the lack of a quadratic Casimir element in \(\mathfrak {p}(n)\otimes \mathfrak {p}(n)\). When M is the trivial representation, the action factors through the Brauer supercategory \(\textit{s}\mathcal {B}{} \textit{r}\). Our main result is an explicit basis theorem for the morphism spaces of 
and, as a consequence, of \(\textit{s}\mathcal {B}{} \textit{r}\). The proof utilises the close connection with the representation theory of \(\mathfrak {p}(n)\). As an application we explicitly describe the centre of all endomorphism algebras, and show that it behaves well under the passage to the associated graded and under deformation.
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Acknowledgements
We thank Gwyn Bellamy, Michael Ehrig, Stephen Griffeth, Joanna Meinel, Travis Schedler and Anne Shepler for helpful discussions. This project was started at the WINART workshop in Banff, and was developed and finalised during several visits of some of the authors to the Hausdorff Center for Mathematics (in particular to MPI and HIM) in Bonn. We thank these places for the excellent working conditions. We are grateful to the referee for useful remarks.
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Balagović, M., Daugherty, Z., Entova-Aizenbud, I. et al. The affine VW supercategory. Sel. Math. New Ser. 26, 20 (2020). https://doi.org/10.1007/s00029-020-0541-4
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DOI: https://doi.org/10.1007/s00029-020-0541-4
Keywords
- Monoidal supercategory
- Lie superalgebras
- Brauer algebras
- Schur–Weyl duality
- Diagram algebras
- Graded and filtered rings