On Computing Mesh Root Systems and the Isotropy Group for Simply-laced Dynkin Diagrams

We continue and complete a Coxeter spectral study (presented in our talk given in SYNASC11, Timisoara, September 2011 [6]) of the root systems in the sense of Bourbaki [4], the mesh geometries Γ(R_Δ, Φ_A) of roots of Δ in the sense of [20], and matrix morsifications A ∈ Mor_Δ, for simply-laced Dynkin diagrams Δ ∈ {A_n, D_n, E₆, E₇, E₈}. One of the main aims of the talk is to present a progress obtained during the last year in solving the problems stated in our talk [6]. We show that the reduced mesh root systems and mesh geometries of roots for each of the Dynkin diagrams Δ can be classified for n ≤ 9 by applying symbolic computer algebra computations and numeric algorithmic computations. We also compute the isotropy group Gl(n, Z)_Δ of Δ defined in [22]-[23], determine its structure and compute the Gl(n, Z)_Δ-orbits of the morsifications A ∈ Mor_Δ, for Δ ∈ {A_n, D_m}, with n ≤ 9 and 4 ≤ m ≤ 8. Results of our computing experiences are presented in three tables of Section 4.