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

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