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

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.