Two Problems in Symmetric Tensor Categories
| dc.contributor.advisor | Ostrik, Victor | |
| dc.contributor.author | Czenky, Agustina | |
| dc.date.accessioned | 2024-08-07T21:30:46Z | |
| dc.date.available | 2024-08-07T21:30:46Z | |
| dc.date.issued | 2024-08-07 | |
| dc.description.abstract | Fix an algebraically closed field k of characteristic $p \geq 0$. A symmetric fusion category $\mathcal C$ over k is a fusion category endowed with a braiding $c_{X,Y}: X\otimes Y \to Y \otimes X$ such that $c_{Y,X}c_{X,Y}=\id_{X\otimes Y} $ for all $X, Y \in \mathcal{C}$. The first part of this dissertation focuses on the study of symmetric fusion categories in positive characteristic. We give lower bounds for the rank of a symmetric fusion category in characteristic $p\geq 5$ in terms of $p$. We also prove that the second Adams operation $\psi_2$ is not the identity for any non-trivial symmetric fusion category, and that symmetric fusion categories satisfying $\psi_2^a=\psi_2^{a-1}$ for some positive integer $a$ are super-Tannakian. As an application, we classify all symmetric fusion categories of rank 3 and those of rank 4 with exactly two self-dual simple objects. The second part of this dissertation treats symmetric categories in the context of topological quantum field theories. We construct a family of unoriented 2-dimensional cobordism theories parametrized by certain triples of sequences, and prove that some specializations of these sequences yield equivalences with an exterior product of Deligne categories. It is known that, modding out the category of 2-dimensional oriented cobordisms by the relation that a handle is the identity, and evaluating 2-spheres to $t$, produces a category equivalent to the Deligne category $\Rep(S_t)$, which generalizes the representation category of the symmetric group $S_n$ from $n\in \mathbb N$ to $t\in \textbf k$. We show an analogous story for unoriented 2-dimensional cobordisms, with a construction that recovers the category $\Rep(S_t\wr \mathbb Z_2)$. This dissertation contains previously published material. | en_US |
| dc.identifier.uri | https://hdl.handle.net/1794/29739 | |
| dc.language.iso | en_US | |
| dc.publisher | University of Oregon | |
| dc.rights | All Rights Reserved. | |
| dc.subject | Fusion categories | en_US |
| dc.subject | Symmetric fusion categories | en_US |
| dc.subject | Topological quantum field theories | en_US |
| dc.title | Two Problems in Symmetric Tensor Categories | |
| dc.type | Electronic Thesis or Dissertation | |
| thesis.degree.discipline | Department of Mathematics | |
| thesis.degree.grantor | University of Oregon | |
| thesis.degree.level | doctoral | |
| thesis.degree.name | Ph.D. |
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