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

We present combinatorial algorithms computing reduced mesh root systems and mesh geometries of roots for the Dynkin diagram Δ=An by means of symbolic computer algebra computations and numeric algorithmic computations in Maple and C++. The problem is reduced to the computation of WΔ-orbits in the set MorΔ of all matrix morsifications A∈Mn(Z) for Δ, their Coxeter polynomials coxA(t), and the Coxeter numbers cA≥2, where WΔ⊆Mn(Z) is the Weyl group of Δ. In case 2≤n is small, a complete classification of WΔ-orbits, is given.