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

Abstract

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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> , Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</sub> ) of roots of Δ in the sense of [20], and matrix morsifications A ∈ Mor <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> , for simply-laced Dynkin diagrams Δ ∈ {A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sub> , E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">7</sub> , E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sub> }. 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) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> of Δ defined in [22]-[23], determine its structure and compute the Gl(n, Z) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> -orbits of the morsifications A ∈ Mor <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Δ</sub> , for Δ ∈ {A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> }, with n ≤ 9 and 4 ≤ m ≤ 8. Results of our computing experiences are presented in three tables of Section 4.