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

22 Sep 2019  ·  Vitale Ryan ·

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. read more

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Representation Theory Combinatorics Rings and Algebras