Affiliations: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland. [email protected], [email protected]
Note: [*] Supported by Polish Research Grant NCN 2011/03/B/ST1/00824
Note: [†] Address for correspondence: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
Abstract: Following the Coxeter spectral analysis of loop-free edge-bipartite graphs Δ and finite posets I, with n ≥ 2 vertices, introduced and developed in [SIAM J. Discrete Math., 27(2013), 827-854], we present a Coxeter spectral classification of finite posets I, with n ≥ 2 elements. Here we study the connected posets I that are non-negative of corank one or two, in the sense that the symmetric Gram matrix 12(CI+CItr)∈𝕄n(ℚ) is positive semi-definite of corank one or two, where CI ∈ 𝕄n(ℤ) is the incidence matrix of I. We study such posets I by means of the Dynkin type DynI and the Coxeter polynomial coxI(t) := det(t · E − CoxI) ∈ ℤ[t], where CoxI := −CI · C−trI ∈ 𝕄n(ℤ) is the Coxeter matrix of I. Among other results, we develop an algorithmic technique that allows us to compute a complete list of such posets I, with |I| ≤ 16, their Dynkin types DynI , and the Coxeter polynomials coxI(t) ∈ ℤ[t]. We prove that, given a pair of such connected posets I and J, the incidence matrices CI and CJ are ℤ-congruent if and only if coxI(t) = coxJ(t) and DynI = DynJ.
Keywords: poset, Coxeter spectrum, Dynkin diagram, Coxeter-Dynkin type
