Diagrammatics for $F_4$

TL;DR

This paper introduces a diagrammatic monoidal category and a functor to modules over the exceptional Lie algebra $F_4$, providing new diagrammatic tools for studying its representation theory.

Contribution

It defines a new diagrammatic framework and establishes a functor to $F_4$ modules, extending classical diagrammatic methods to exceptional Lie algebras.

Findings

Established a full and essentially surjective monoidal functor to $F_4$ modules

Created diagrammatic tools analogous to Brauer categories for $F_4$

Facilitated new approaches to $F_4$ representation theory

Abstract

We define a diagrammatic monoidal category, together with a full and essentially surjective monoidal functor from this category to the category of modules over the exceptional Lie algebra of type $F_4$. In this way, we obtain a set of diagrammatic tools for studying type $F_4$ representation theory that are analogous to those of the oriented and unoriented Brauer categories in classical type.