Diagram categories and invariant theory for classical groups and supergroups

TL;DR

This paper introduces diagram categories to study invariant theory of classical groups and supergroups, providing new fundamental theorems and algebraic presentations using diagrammatic methods.

Contribution

It develops the concept of diagram categories and applies them to establish fundamental theorems and algebraic presentations for classical supergroups and related structures.

Findings

First and second fundamental theorems for classical supergroups

Presentation of endomorphism algebras for orthogonal and symplectic groups

Extension of invariant theory to supergroups and diagrammatic methods

Abstract

We introduce the notion of a diagram category and discuss its application to the invariant theory of classical groups and super groups, with some indications concerning extensions to quantum groups and quantum super groups. Tensor functors from various diagram categories to categories of representnations are introduced and their properties investigated, leading to first and second fundamental theorems of invariant theory for classical super groups, which include the classical groups as special cases. Application of diagrammatic methods enables the constructionof a presentation for endomorphism algebras for te orthogonal and symplectic groups, leading to the solution ofproblems raised by the work of Brauer and Weyl.