A Framework for Coxeter Spectral Analysis of Edge-bipartite Graphs, their Rational Morsifications and Mesh Geometries of Root Orbits

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 coxΔt ∈ $\mathbb{Z}$[t] and the $\mathbb{Z}$-bilinear Gram form bΔ : $\mathbb{Z}$n × $\mathbb{Z}$n → $\mathbb{Z}$ of Δ [SIAM J. Discrete Math. 272013]. 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 Gln,$\mathbb{Z}$Δ of Gln, $\mathbb{Z}$ 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 Gln, $\mathbb{Z}$Δ-invariant. It is shown that, for n ≤ 6, speccΔ is the spectrum of one of the Coxeter polynomials listed in Tables 3.11-3.11a 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.