Diagrammatics for real supergroups

TL;DR

This paper introduces diagrammatic supercategories generalizing Brauer categories, establishing their connections to supergroups and deriving fundamental invariant theory results for various real supergroups.

Contribution

It develops new diagrammatic supercategories linked to supergroups and proves their fullness, leading to fundamental invariant theory theorems for real forms of complex supergroups.

Findings

Supercategories generalize oriented and unoriented Brauer categories.

Superfunctors are full for central real division superalgebras.

Derived first fundamental theorems for real supergroup forms.

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.