Date of Award
5-1-2024
Language
English
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
College/School/Department
Department of Mathematics and Statistics
Dissertation/Thesis Chair
Rongwei Yang
Committee Members
Matthew Zaremsky, Antun Milas, Michael Stessin
Keywords
braid groups, characteristic polynomial, Coxeter groups, point groups, projections
Subject Categories
Physical Sciences and Mathematics
Abstract
This dissertation examines some applications of the characteristic polynomial \[Q_A(z):=\det (z_0I+z_1A_1+\cdots+z_nA_n),\] where $A_0,\dots,A_n$ are complex square matrices and $(z_0,\dots,z_n)\in\mathbb{C}^{n+1}$. In particular, it proves that a Coxeter system is completely determined by its characteristic polynomial with respect to the Tits representation and that the characteristic polynomial for pairs of projection matrices can be calculated using their orthogonal decomposition described by Halmos~\cite{Hal}. A consequence of this calculation is that their characteristic polynomial is a complete unitary invariant for the pairs of projections, which may be alternatively proven by their trace. These results have been published in~\cite{HY}. The unitary equivalence of tuples of projections, the spectral equivalence of axial point groups, and the irreducibility of the characteristic polynomial of the braid group $B_4$ with respect to the Burau representation are also considered.
Recommended Citation
Howell, Kate Helen Auguste, "The Characteristic Polynomial And Its Applications" (2024). Legacy Theses & Dissertations (2009 - 2024). 3323.
https://scholarsarchive.library.albany.edu/legacy-etd/3323
