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 𝒰ℬigr_n of loop-free edge-bipartite (signed) graphs Δ, with n ≥ 2 vertices, by means of the Coxeter number c_Δ, the Coxeter spectrum specc_Δ of Δ, that is, the spectrum of the Coxeter polynomial cox_Δ(t) ∈ $\mathbb{Z}$[t] and the $\mathbb{Z}$-bilinear Gram form b_Δ : $\mathbb{Z}$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 𝒰ℬigr_n reduces to the Coxeter spectral classification of rational matrix morsifications A ∈ $\widehat{M}$or_DΔ for a simply-laced Dynkin diagram DΔ associated with Δ. Given Δ in 𝒰ℬigr_n, 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 $\widehat{M}$or_Δ of rational matrix morsifications A of Δ in such a way that the map A $\mapsto$ (specc_A, det A, c_Δ) is Gl(n, $\mathbb{Z}$)_Δ-invariant. It is shown that, for n ≤ 6, specc_Δ 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 𝒰ℬigr_n, with specc_Δ = specc_Δ′, are $\mathbb{Z}$-bilinear equivalent, is studied in the paper. The problem if any $\mathbb{Z}$-invertible matrix A ∈ M_n($\mathbb{Z}$) is $\mathbb{Z}$-congruent with its transpose A^tr is also discussed.