Planar algebra presentations of $\text{URep}_{\mathbb{C}}(\mathbb{C}^+)$ and $\text{URep}_{\mathbb{F}_p}(\mathbb{F}_p^+)$

TL;DR

This paper constructs explicit planar algebra presentations for unipotent representations of additive groups over complex numbers and finite fields, revealing new generators and paving the way for future invariant theory research.

Contribution

It introduces novel planar algebra presentations for unipotent groups over different fields, including new generators for characteristic p cases, using jellyfish and light leaf techniques.

Findings

Presented planar algebra descriptions for $ ext{URep}_{ ext{C}}( ext{C}^+)$ and $ ext{URep}_{ ext{F}_p}( ext{F}_p^+)$

Discovered new generators in large box spaces as p increases

Outlined future research directions in invariant theory and fundamental theorems

Abstract

We give presentations of the planar algebra of unipotent representations of the groups $\mathbb{C}$ and $\mathbb{F}_p$ under addition using jellyfish and light leaf style arguments. These are some of the most natural examples of non-semisimple planar algebras. For the characteristic $p$ family of examples, a new generator appears in arbitrarily large box spaces as $p$ increases. We point toward future directions in getting results on first and second fundamental theorems for rings of vector invariants, as well as generalization of the examples given.