Diagrammatics for real supergroups

Samchuck-Schnarch, Saima; Savage, Alistair

doi:10.5802/art.7

Samchuck-Schnarch, Saima ¹; Savage, Alistair ¹

¹ Department of Mathematics and Statistics, University of Ottawa STEM Building, 150 Louis-Pasteur Ottawa ON K1N 6N5 Canada

Annals of Representation Theory, Volume 1 (2024) no. 2, pp. 125-191.

Abstract

We introduce two families of diagrammatic monoidal supercategories. The first family, depending on an associative superalgebra, generalizes the oriented Brauer category. The second, depending on an involutive superalgebra, generalizes the unoriented Brauer category. These two families of supercategories admit natural superfunctors to supercategories of supermodules over general linear supergroups and supergroups preserving superhermitian forms, respectively. We show that these superfunctors are full when the superalgebra is a central real division superalgebra. As a consequence, we obtain first fundamental theorems of invariant theory for all real forms of the general linear, orthosymplectic, periplectic, and isomeric supergroups. We also deduce equivalences between monoidal supercategories of tensor supermodules over the real forms of a complex supergroup.

Received: 2023-01-11
Revised: 2024-01-03
Accepted: 2024-01-31
Published online: 2024-05-31

DOI: 10.5802/art.7

Classification: 18M05, 18M30, 17B10, 18M25
Keywords: Monoidal category, supercategory, supergroup, string diagram, invariant theory, Deligne category, interpolating category

Author's affiliations:

Samchuck-Schnarch, Saima ¹; Savage, Alistair ¹

¹ Department of Mathematics and Statistics, University of Ottawa STEM Building, 150 Louis-Pasteur Ottawa ON K1N 6N5 Canada

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Samchuck-Schnarch, Saima; Savage, Alistair. Diagrammatics for real supergroups. Annals of Representation Theory, Volume 1 (2024) no. 2, pp. 125-191. doi : 10.5802/art.7. https://art.centre-mersenne.org/articles/10.5802/art.7/

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