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

Abstract EN

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 J≡(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 posets J such that the symmetric Gram matrix GJ:=(1/2)[CJ+CJtr]∈?J(ℚ) is positive semidefinite, where CJ∈?J(ℤ) is the incidence matrix of J.

Following the idea of Drozd mentioned earlier, we associate to J its Coxeter matrix CoxJ:=-CJ·CJ-tr, its Coxeter spectrum speccJ, a Coxeter polynomial coxJ(t)∈ℤ[t], and a Coxeter number cJ.

In case GJ is positive semi-definite, we also associate to J a reduced Coxeter number čJ, and the defect homomorphism ∂J:ℤJ→ℤ.

In this case, the Coxeter spectrum speccJ is a subset of the unit circle and consists of roots of unity.

In case GJ 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∈{?̃n,?̃6,?̃7,?̃8} associated with J.

Our aim of the Coxeter spectral analysis of such posets J is to answer the question when the Coxeter type CtypeJ:=(speccJ,cJ, čJ) of J determines its incidence matrix CJ (and, hence, the poset J) uniquely, up to a ℤ-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 ℤ-invertible matrix A∈?n(ℤ), there is B∈?n(ℤ) such that Atr=Btr·A·B and B2=E is the identity matrix.