International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 743734, 22 pages
Research Article

A Framework for Coxeter Spectral Classification of Finite Posets and Their Mesh Geometries of Roots

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Ulica Chopina 12/18, 87-100 Toruń, Poland

Received 28 March 2012; Accepted 1 November 2012

Academic Editor: Marco Squassina

Copyright © 2013 Daniel Simson and Katarzyna Zając. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Following our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posets . One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by Drozd [Funct. Anal. Appl. 8(1974), 219–225]. We are mainly interested in a Coxeter spectral classification of posets such that the symmetric Gram matrix is positive semidefinite, where is the incidence matrix of . Following the idea of Drozd mentioned earlier, we associate to its Coxeter matrix , its Coxeter spectrum , a Coxeter polynomial , and a Coxeter number . In case is positive semi-definite, we also associate to a reduced Coxeter number , and the defect homomorphism . In this case, the Coxeter spectrum is a subset of the unit circle and consists of roots of unity. In case is positive semi-definite of corank one, we relate the Coxeter spectral properties of the posets with the Coxeter spectral properties of a simply laced Euclidean diagram associated with . Our aim of the Coxeter spectral analysis of such posets is to answer the question when the Coxeter type of determines its incidence matrix (and, hence, the poset ) uniquely, up to a -congruency. In connection with this question, we also discuss the problem studied by Horn and Sergeichuk [Linear Algebra Appl. 389(2004), 347–353], if for any -invertible matrix , there is such that and is the identity matrix.