Computer Algebra Technique for Coxeter Spectral Study of Edge-bipartite Graphs and Matrix Morsifications of Dynkin Type $\mathbb{A}_n$

Abstract

By applying computer algebra tools (mainly, Maple and C++), given the Dynkin diagram $\Delta = \mathbb{A}_n$ , with n ≥ 2 vertices and the Euler quadratic form $q_\Delta : \mathbb{Z}^n \rightarrow \mathbb{Z}$ , we study the problem of classifying mesh root systems and the mesh geometries of roots of Δ (see Section 1 for details). The problem reduces to the computation of the Weyl orbits in the set $Mor_\Delta \subseteq \mathbb{M}_n(\mathbb{Z})$ of all matrix morsifications A of q_Δ, i.e., the non-singular matrices $A \in \mathbb{M}_n(\mathbb{Z})$ such that (i) q_Δ(v) = v · A · v^tr, for all $v \in \mathbb{Z}^n$ , and (ii) the Coxeter matrix Cox_A := −A · A^−tr lies in $Gl(n,\mathbb{Z})$ . The Weyl group $\mathbb{W}_\Delta \subseteq Gl(n, \mathbb{Z})$ acts on Mor_Δ and the determinant det $A \in \mathbb{Z}$ , the order c_A ≥ 2 of Cox_A (i.e. the Coxeter number), and the Coxeter polynomial $cox_A(t) := det(t \centerdot E \minus Cox_A) \in \mathbb{Z}[t]$ are $\mathbb{W}_\Delta$ -invariant. The problem of determining the $\mathbb{W}_\Delta$ -orbits $\cal{O}rb(A)$ of Mor_Δ and the Coxeter polynomials cox_A(t), with $A \in Mor_\Delta$ , is studied in the paper and we get its solution for n ≤ 8, and $A = [a_{ij}] \in Mor_{\mathbb{A}}_n$ , with $\vert a_{ij} \vert \le 1$ . In this case, we prove that the number of the $\mathbb{W}_\Delta$ -orbits $\cal{O}rb(A)$ and the number of the Coxeter polynomials cox_A(t) equals two or three, and the following three conditions are equivalent: (i) $\cal{O}rb(A) = \mathbb{O}rb(A\prime)$ , (ii) cox_A(t) = cox_A′(t), (iii) c_A · det A = c_A′ · det A′. We also construct: (a) three pairwise different $\mathbb{W}_\Delta$ -orbits in Mor_Δ, with pairwise different Coxeter polynomials, if $\Delta = \mathbb{A}_{2m \minus 1}$ and m ≥ 3; and (b) two pairwise different $\mathbb{W}_\Delta$ -orbits in Mor_Δ, with pairwise different Coxeter polynomials, if $\Delta = \mathbb{A}_{2m}$ and m ≥ 1.