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

Abstract

Following our paper [Linear Algebra Appl. 433(2010), 699–717], we present a framework and computational tools for the Coxeter spectral classification of finite posetsJ≡(J,⪯). 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 posetsJsuch that the symmetric Gram matrixGJ:=(1/2)[CJ+CJtr]∈𝕄J(ℚ)is positive semidefinite, whereCJ∈𝕄J(ℤ)is the incidence matrix ofJ. Following the idea of Drozd mentioned earlier, we associate toJits Coxeter matrixCoxJ:=-CJ·CJ-tr, its Coxeter spectrumspeccJ, a Coxeter polynomialcoxJ(t)∈ℤ[t], and a Coxeter number  cJ. In caseGJis positive semi-definite, we also associate toJa reduced Coxeter number  čJ, and the defect homomorphism∂J:ℤJ→ℤ. In this case, the Coxeter spectrumspeccJis a subset of the unit circle and consists of roots of unity. In caseGJis positive semi-definite of corank one, we relate the Coxeter spectral properties of the posetsJwith the Coxeter spectral properties of a simply laced Euclidean diagramDJ∈{𝔻̃n,𝔼̃6,𝔼̃7,𝔼̃8}associated withJ. Our aim of the Coxeter spectral analysis of such posetsJis to answer the question when the Coxeter typeCtypeJ:=(speccJ,cJ,  čJ)ofJdetermines its incidence matrixCJ(and, hence, the posetJ) uniquely, up to aℤ-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 anyℤ-invertible matrixA∈𝕄n(ℤ), there isB∈𝕄n(ℤ)such thatAtr=Btr·A·BandB2=Eis the identity matrix.