A framework for Coxeter spectral classification of finite posets and their mesh geometries of roots.
type: Item
instance of: scholarly article
title: A framework for Coxeter spectral classification of finite posets and their mesh geometries of roots.
zbMATH Open document ID: 1284.16018
DOI: 10.1155/2013/743734
author: Daniel Simson
author: Katarzyna Zając
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2013-05-23T00:00:00Z
review text: Summary: Following our paper [\textit{D. Simson}, Linear Algebra Appl. 433, No. 4, 699-717 (2010; Zbl 1196.15022)], we present a framework and computational tools for the Coxeter spectral classification of finite posets \(J\equiv(J,\preceq)\). One of the main motivations for the study is an application of matrix representations of posets in representation theory explained by \textit{Yu. A. Drozd} [Funct. Anal. Appl. 8, 219-225 (1974); translation from Funkts. Anal. Prilozh. 8, No. 3, 34-42 (1974; Zbl 0356.06003)]. We are mainly interested in a Coxeter spectral classification of posets \(J\) such that the symmetric Gram matrix \(G_J:=(1/2)[C_J+C_J^{\text{tr}}]\in\mathbb M_J(\mathbb Q)\) is positive semidefinite, where \(C_J\in\mathbb M_J(\mathbb Z)\) is the incidence matrix of \(J\). Following the idea of Drozd mentioned earlier, we associate to \(J\) its Coxeter matrix \(\text{Cox}_J:=-C_J\cdot C^{-\text{tr}}_J\), its Coxeter spectrum \(\mathbf{specc}_J\), a Coxeter polynomial \(\text{cox}_J(t)\in\mathbb Z[t]\), and a Coxeter number \(\mathbf c_J\). In case \(G_J\) is positive semi-definite, we also associate to \(J\) a reduced Coxeter number \(\check{\mathbf c}_J\), and the defect homomorphism \(\partial_J\colon\mathbb Z^J\to\mathbb Z\). In this case, the Coxeter spectrum \(\mathbf{specc}_J\) is a subset of the unit circle and consists of roots of unity. In case \(G_J\) is positive semi-definite of corank one, we relate the Coxeter spectral properties of the posets \(J\) with the Coxeter spectral properties of a simply laced Euclidean diagram \(DJ\in\{\widetilde{\mathbb D}_n,\widetilde{\mathbb E}_6,\widetilde{\mathbb E}_7,\widetilde{\mathbb E}_8\}\) associated with \(J\). Our aim of the Coxeter spectral analysis of such posets \(J\) is to answer the question when the Coxeter type \(\mathbf{Ctype}_J:=(\mathbf{specc}_J,\mathbf c_J,\check{\mathbf c}_J)\) of \(J\) determines its incidence matrix \(C_J\) (and, hence, the poset \(J\)) uniquely, up to a \(\mathbb Z\)-cong…
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 06A07
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 06A11
zbMATH DE Number: 6165369
zbMATH Keywords: classification of finite posets
zbMATH Keywords: matrix representations of posets
zbMATH Keywords: positive semidefinite symmetric Gram matrices
zbMATH Keywords: Coxeter matrices
zbMATH Keywords: Coxeter spectra
zbMATH Keywords: Coxeter polynomials
zbMATH Keywords: Coxeter numbers
describes a project that uses: Maple
describes a project that uses: Python
MaRDI profile type: Publication
Wikidata QID: Q58988767
full work available at URL: https://doi.org/10.1155/2013/743734
OpenAlex ID: W2016930771
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description: scientific article; zbMATH DE number 6165369