Year
2025
Season
Fall
Paper Type
Master's Thesis
College
College of Arts and Sciences
Degree Name
Master of Science in Mathematical Sciences (MS)
Department
Mathematics & Statistics
NACO controlled Corporate Body
University of North Florida. Department of Mathematics and Statistics
Committee Chairperson
Dr. Allan Merino
Second Advisor
Dr. Jose Franco
Third Advisor
Dr. Daniela Genova
Department Chair
Dr. Richard Patterson
Abstract
The classical invariant theory of matrix groups plays a fundamental role in modern algebra and geometry, with origins in Hilbert’s Fourteenth Problem concerning the finite generation of rings of invariants. In this thesis, we study polynomial functions on spaces of matrices and investigate their invariants under natural actions of classical groups. After recalling the geometric and algebraic motivations behind Hilbert’s question, we focus on invariants for several complex Lie groups, including the general linear group GL(V), the symplectic group Sp(V), the orthogonal group O(V), and the special orthogonal group SO(V). Building on foundational work of Hermann Weyl and a modern treatment by Claudio Procesi, we present an explicit collection of polynomial invariants. We then review classical results such as the First and Second Fundamental Theorems of invariant theory through concrete examples, illustrating how trace polynomials, Pfaffians, and their symplectic analogues arise naturally. The talk concludes with a selection of applications, including connections to algebraic geometry and the study of character varieties.
Suggested Citation
Diller, John, "Polynomial invariants for classical invariant groups" (2025). UNF Graduate Theses and Dissertations. 1387.
https://digitalcommons.unf.edu/etd/1387