Principal Elements of Mixed-Sign Coxeter Systems

• Other Authors: Armstrong, Johnathon Kyle (authoraut)
• Format: Others
• Language: English; English
• Published: Florida State University
• Subjects: Mathematics
• Online Access: http://purl.flvc.org/fsu/fd/FSU_migr_etd-4697

In this thesis we generalize results from classical Coxeter systems to mixed-sign Coxeter systems which are denoted by a triple (W,S,B)consisting of a reflection group W, a distinguished set of generators Sfor the group for W, and a bilinear form Bon R n. A generator s i in the set S is defined to negate the ith basis vector of R n and fix the set of vectors v which are orthogonal relative to B. Classical Coxeter theory works in this fashion, here we generalize this notion to encompass both Coxeter systems in addition to mixed-sign Coxeter systems. As in classical Coxeter theory, we show that the bilinear form may be used to compute an element of the reflection group called a principal element. In classical Coxeter groups, the principal elements have been shown to have special properties. The so-called deletion condition is a property of classical Coxeter systems which allows Coxeter groups to have a presentation which only depends on pairwise relationships between generators. Here, we show that mixed-sign Coxeter systems do not generally have the deletion condition. We give a correspondence between a graph $\Gamma$ and the reflection system (W,S,B). We refer to the reflection group associated to &Gamma by W (&Gamma). We show an isomorphism of mixed-sign Coxeter groups; explicitly if &Gamma is a bipartite mixed-sign Coxeter graph and &Gamma is the mixed-sign Coxeter graph with all the nodes of &Gamma- negated then (W,S,B(&Gamma)) and (W,S,B(&Gamma-)) are conjugate reflection systems. Furthermore, we indicate the the bipartite condition is necessary. We show a class of examples; odd cycles with all negative nodes where negating all the nodes gives a reflection system which is not conjugate. Additionally, we show that the spectral radius of mixed-sign Coxeter elements are not bounded below by the bipartite eigenvalue of the mixed-sign Coxeter system, this is another distinguishing feature of mixed-sign Coxeter systems from their classical counterparts and provides an interesting avenue of research to pursue in the future. === A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. === Spring Semester, 2012. === December 2, 2011. === Coxeter Systems, graph, mixed-sign Coxeter, polynomial, reflection groups, spectral radius === Includes bibliographical references. === Eriko Hironaka, Professor Directing Dissertation; Kathleen Petersen, Professor Co-Directing Dissertation; Eric Chicken, University Representative; Ettore Aldrovandi, Committee Member; Steven Bellenot, Committee Member; Mark van Hoeij, Committee Member.