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 qΔ, i.e., the non-singular matrices
$A \in \mathbb{M}_n(\mathbb{Z})$
such that (i) qΔ(v) = v · A · vtr, for all
$v \in \mathbb{Z}^n$
, and (ii) the Coxeter matrix CoxA := −A · A−tr lies in
$Gl(n,\mathbb{Z})$
. The Weyl group
$\mathbb{W}_\Delta \subseteq Gl(n, \mathbb{Z})$
acts on MorΔ and the determinant det
$A \in \mathbb{Z}$
, the order cA ≥ 2 of CoxA (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 MorΔ and the Coxeter polynomials coxA(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 coxA(t) equals two or three, and the following three conditions are equivalent: (i)
$\cal{O}rb(A) = \mathbb{O}rb(A\prime)$
, (ii) coxA(t) = coxA′(t), (iii) cA · det A = cA′ · det A′. We also construct: (a) three pairwise different
$\mathbb{W}_\Delta$
-orbits in MorΔ, 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 MorΔ, with pairwise different Coxeter polynomials, if
$\Delta = \mathbb{A}_{2m}$
and m ≥ 1.
