Date on Honors Thesis

5-2022

Major

Mathematics

Minor

Computer Science

Examining Committee Member

Dr. Robert G. Donnelly, Advisor

Examining Committee Member

Dr. Elizabeth A. Donovan, Committee Member

Examining Committee Member

Dr. Timothy Schroeder, Committee Member

Abstract/Description

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 the type D_n QDPs and novel derivations of the QDPs of types A_n, B_n, C_n, and D_n are provided using an inductive proof through the Networked-Numbers Game. This provides a combinatorial avenue of approach to a topic traditionally only attainable through Lie theory.

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