Affiliations: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland. [email protected]
Note: [] Supported by Polish Research Grant NCN 2011/03/B/ST1/00824 Address for correspondence: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
Abstract: Following the spectral Coxeter analysis of matrix morsifications for Dynkin diagrams, the spectral graph theory, a graph coloring technique, and algebraic methods in graph theory, we continue our study of the category 𝒰ℬigrn of loop-free edge-bipartite (signed) graphs Δ, with n ≥ 2 vertices, by means of the Coxeter number cΔ, the Coxeter spectrum speccΔ of Δ, that is, the spectrum of the Coxeter polynomial coxΔ(t) ∈ $\mathbb{Z}$[t] and the $\mathbb{Z}$-bilinear Gram form bΔ : $\mathbb{Z}$n × $\mathbb{Z}$n → $\mathbb{Z}$ of Δ [SIAM J. Discrete Math. 27(2013)]. Our main inspiration for the study comes from the representation theory of posets, groups and algebras, Lie theory, and Diophantine geometry problems. We show that the Coxeter spectral classification of connected edge-bipartite graphs Δ in 𝒰ℬigrn reduces to the Coxeter spectral classification of rational matrix morsifications A ∈ $\widehat{M}$orDΔ for a simply-laced Dynkin diagram DΔ associated with Δ. Given Δ in 𝒰ℬigrn, we study the isotropy subgroup Gl(n,$\mathbb{Z}$)Δ of Gl(n, $\mathbb{Z}$) that contains the Weyl group $\mathbb{W}$Δ and acts on the set $\widehat{M}$orΔ of rational matrix morsifications A of Δ in such a way that the map A $\mapsto$ (speccA, det A, cΔ) is Gl(n, $\mathbb{Z}$)Δ-invariant. It is shown that, for n ≤ 6, speccΔ is the spectrum of one of the Coxeter polynomials listed in Tables 3.11-3.11(a) (we determine them by computer search using symbolic and numeric computation). The question, if two connected positive edge-bipartite graphs Δ,Δ′ in 𝒰ℬigrn, with speccΔ = speccΔ′, are $\mathbb{Z}$-bilinear equivalent, is studied in the paper. The problem if any $\mathbb{Z}$-invertible matrix A ∈ Mn($\mathbb{Z}$) is $\mathbb{Z}$-congruent with its transpose Atr is also discussed.
