dc.contributor.advisor | Kujawa, Jonathan | |
dc.contributor.author | Reynolds, Ryan | |
dc.date.accessioned | 2023-05-17T14:37:48Z | |
dc.date.available | 2023-05-17T14:37:48Z | |
dc.date.issued | 2023-05-12 | |
dc.identifier.uri | https://hdl.handle.net/11244/337707 | |
dc.description.abstract | There is a categorical equivalence between the Temperley--Lieb category $TL(2)$ and the full subcategory of $SU(2)$-\textbf{mod} with objects given by $V^{\otimes k}$ where $V$ is the tautological $SU(2)$-module and $k$ is a non-negative integer. The first results in this dissertation develop new diagrammatic categories which are shown to be equivalent to similarly defined full subcategories of $G$-\textbf{mod} for certain finite subgroups $G$ of $SU(2)$. The diagrams which generate the Temperley--Lieb category are shown to be linear combinations of the generating diagrams for these newly defined diagrammatic categories. The main result of this paper utilizes the representation graph of a group $G$, $R(V,G)$, and gives a general construction of a diagrammatic category $\mathbf{Dgrams}_{R(V,G)}$. The proof of the main theorem shows that, given explicit criteria, there is an equivalence of categories between a quotient category of $\mathbf{Dgrams}_{R(V,G)}$ and a full subcategory of $G-\textbf{mod}$ with objects being the tensor products of finitely many irreducible $G$-modules. | en_US |
dc.language | en_US | en_US |
dc.subject | Mathematics. | en_US |
dc.subject | Representation Theory | en_US |
dc.subject | Combinatorics | en_US |
dc.subject | Category Theory | en_US |
dc.subject | Diagrammatic Categories | en_US |
dc.subject | McKay Graphs | en_US |
dc.subject | Representation Graphs | en_US |
dc.subject | Algebra | en_US |
dc.subject | McKay Correspondence | en_US |
dc.subject | Temperley-Lieb | en_US |
dc.subject | Special Unitary Group | en_US |
dc.subject | Semisimple category | en_US |
dc.subject | Monoidal category | en_US |
dc.subject | k-linear category | en_US |
dc.title | Diagrammatic Categories which arise from Directed Graphs | en_US |
dc.contributor.committeeMember | Docampo-Alvarez, Roi | |
dc.contributor.committeeMember | Lifschitz, Lucy | |
dc.contributor.committeeMember | Muller, Greg | |
dc.contributor.committeeMember | Chappell, David | |
dc.date.manuscript | 2023-05-05 | |
dc.thesis.degree | Ph.D. | en_US |
ou.group | Dodge Family College of Arts and Sciences::Department of Mathematics | en_US |
shareok.orcid | 0000-0002-2850-0600 | en_US |