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

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. 1. Introduction In the present paper, we continue our Coxeter spectral study of finite posets, started in [1], in a close connection with the Coxeter spectral technique introduced in [2–4] for acyclic edge-bipartite graphs or signed graphs in the sense of [5]. We also follow some of the techniques of representation theory, graph combinatorics, and the spectral graph theory; see [6–31]. Here, we use the terminology and notation introduced in [1, 4, 26–28]. We denote by the set of nonnegative integers, the ring of integers, and the rational number field. Given , we view as a free abelian group and denote by the standard -basis of . Given an index set , we denote by the abelian group of all vectors , with integer coordinates , by the -algebra of all square by integral matrices, and by the identity matrix. In particular, , with , is the -algebra of all square by matrices. The group is called the (integral) general linear group. We say that two square rational matrices are -equivalent, or -congruent, (and denote ) if there is a matrix such that . By a poset we mean a finite partially ordered set with