Mathematics Commons^™

Quantum Dimension Polynomials: A Networked-Numbers Game Approach, Nicholas Gaubatz

Honors College Theses

The Networked-Numbers Game--a mathematical "game'' played on a simple graph--is incredibly accessible and yet surprisingly rich in content. The Game is known to contain deep connections to the finite-dimensional simple Lie algebras over the complex numbers. On the other hand, Quantum Dimension Polynomials (QDPs)--enumerative expressions traditionally understood through root systems--corresponding to the above Lie algebras are complicated to derive and often inaccessible to undergraduates. In this thesis, the Networked-Numbers Game is defined and some known properties are presented. Next, the significance of the QDPs as a method to count combinatorially interesting structures is relayed. Ultimately, a novel closed-form expression of …

Finite Index Right-Angled Mock Artin Groups In Right-Angled Mock Reflection Groups, Zachary Marcum

Murray State Theses and Dissertations

Associated to any graph Γ are several groups where their presentations are encoded by Γ. Two such groups are right-angled Coxeter groups and right-angled Artin groups. By introducing more structure to Γ, what we call "local involutions," the graph Γ becomes a right-angled mock reflection system and encodes the presentation of two more groups: right-angled mock reflection groups and right-angled mock Artin groups.

Here we show that every right-angled mock Artin group associated with an n-gon graph with local involutions is a finite index subgroup in some right-angled mock reflection group. We employ a strategy similar to the one Davis …