On combinatorial algorithms computing mesh root systems and matrix morsifications for the Dynkin diagram An

On Coxeter Type Classification of Loop-Free Edge-Bipartite Graphs and Matrix Morsifications

“…Our main aim in this section is to show that the computation of the set CGpol + n of all polynomials cox Δ (t), with positive edge-bipartite graphs Δ in UBigr n , n ≥ 2, and the proof of Theorem 2.3 reduces to the computation of Gl(n, Z) D -orbits on the set Mor D of matrix morsifications for simply laced Dynkin diagrams D. Using this idea, we construct algorithmic procedures for the Coxeter spectral analysis of loop-free edge-bipartite graphs in UBigr n , by applying symbolic and numerical computations in Linux, Maple and C++, with GNU Scientific Library. Here we mainly apply the technique and results given in [8]- [10], [15]- [16], [19]- [22], and [25]- [27].…”

“…Module categories and their derived categories D b (R) are studied by means of their AR-quivers and Φ-mesh root systems in the sense of [24], i.e., root systems R ⊆ Z n equipped with a Φ-mesh geometry structure Γ(R, Φ) defined by a non-trivial group automorphism Φ : Z n → Z n . This means that the root system R is a Φ-invariant subset of Z n , and the Φ-orbits in R form a Φ-translation quiver (oriented graph) consisting of Φ-meshes, see [10] and [24] (see also [1], [22], [32] for details).…”

“…We recall from [3]- [13] and [24]- [27] that a motivation for the study of mesh root systems, mesh geometries of roots, matrix morsifications, and their connections with some Diophantine geometry problems and the Hilbert's tenth problem (see [18]) is given in [10] and [24]- [27]. We recall that root systems and Dynkin diagrams play an important role in several branches of mathematics including Lie theory, algebraic geometry, group theory, geometry, singularity theory, and representation theory, see [1], [27], [32].…”