Affiliations: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland. [email protected]
Note: [] The research is supported by Polish Research Grant NCN 2011/03/B/ST1/00824 Address for correspondence: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
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 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.
