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A Horizontal Mesh Algorithm for a Class of Edge-bipartite Graphs and their Matrix Morsifications

Kaniecki M., Kosakowska J., Malicki P., Marczak G.

Fundamenta Informaticae

2015

Vol. 136, nr 4

345--379

We construct a horizontal mesh algorithm for a study of a special type of mesh root systems of connected positive loop-free edge-bipartite graphs Δ, with n ≥ 2 vertices, in the sense of [SIAM J. Discrete Math. 27 (2013), 827–854] and [Fund. Inform. 124 (2013), 309-338]. Given such a loop-free edge-bipartite graph Δ, with the non-symmetric Gram matrix ˇGΔ ∈ Mn(Z) and the Coxeter transformation ΦA : Zn → Zn defined by a quasi-triangular matrix morsification A ∈ Mn(Z) of Δ satisfying a non-cycle condition, our combinatorial algorithm constructs a ΦA-mesh root system structure Γ(RΔ,ΦA) on the finite set of all ΦA-orbits of the irreducible root system RΔ := {v ∈ Zn; v · ˇGΔ · vtr = 1}. We apply the algorithm to a graphical construction of a ΦI - mesh root system structure Γ(RI ,ΦI ) on the finite set of ΦI -orbits of roots of any poset I with positive definite Tits quadratic form bqI : ZI → Z.

Algorithms Determining Matrix Morsifications,Weyl orbits, Coxeter Polynomials and Mesh Geometries of Roots for Dynkin Diagrams

Simson D.

Fundamenta Informaticae

2013

Vol. 123, nr 4

447--490

By computer algebra technique and computer computations, we solve the mesh morsification problems 1.10 and present a classification of irreducible mesh roots systems, for some of the simply-laced Dynkin diagramsΔ ∈ {An,Dn, E6, E7,E8}. The methods we use show an importance of computer algebra tools in solving difficult modern algebra problems of enough high complexity that had no solution by means of standard theoretical tools only. Inspired by results of Sato [Linear Algebra Appl. 406(2005), 99-108] and a mesh quiver description of indecomposable representations of finite-dimensional algebras and their derived categories explained in [London Math. Soc. Lecture Notes Series, Vol. 119, 1988] and [Fund. Inform. 109(2011), 425-462] (see also 5.11), given a Dynkin diagram Δ, with n vertices and the Euler quadratic form qΔ : Zn → Z, we study the set MorΔ ⊆ Mn(Z) of all morsifications of qΔ [37], i.e., the non-singular matrices A ∈ Mn(Z) such that its Coxeter matrix CoxA := −A · A−tr lies in Gl(n, Z) and qΔ(v) = v · A · vtr, for all v ∈ Zn. The matrixWeyl groupWΔ (2.13) acts on MorΔ and the determinant detA ∈ Z, the order cA ≥ 2 of CoxA (i.e. the Coxeter number), and the Coxeter polynomial coxA(t) := det(t ·E−CoxA) ∈ Z[t] are WΔ-invariant. Moreover, the finite set RqΔ = {v ∈ Zn; qΔ(v) = 1} of roots of qΔ is CoxA- invariant. The following problems are studied in the paper: (a) determine the WΔ-orbits Orb(A) of MorΔ and the set CPolΔ = {coxA(t); A ∈ MorΔ}, (b) construct a finite minimal CoxA-mesh quiver in Zn containing all CoxA-orbits of the finite set RqΔ of roots of qΔ. We prove that CPolΔ is a finite set and we construct algorithms allowing us to solve the problems for the morsifications A = [aij ] ∈ MorΔ, with |aij | ≤ 2. In this case, by computer algebra technique and computer computations, we prove that, for n ≤ 8, the number of the WΔ-orbits Orb(A) is at most 6, sΔ := |CPolΔ| ≤ 9 and, given A,A′ ∈ MorΔ and n ≤ 7, the following three conditions are equivalent: (i) A′ = Btr · A · B, for some B ∈ Gl(n, Z), (ii) coxA(t) = coxA′ (t), and (iii) cA · detA = cA′ · detA′. We also show that sΔ equals 6, 5, and 9, if Δ is the diagram E6, E7, and E8, respectively.