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.

References in corpus (4)

• Light ladders and clasp conjectures
• Modular representation theory in type A via Soergel bimodules
• A Formula for the Jones-Wenzl Projections
• Skein theory for the ADE planar algebras