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Graph Theoretical and Algorithmic Characterizations of Positive Definite Symmetric Quasi-Cartan Matrices

Abarca M., Rivera D.

Fundamenta Informaticae

2016

Vol. 149, nr 3

241--261

A well known constructive proof for the ADE-classification of many mathematical objects, such as positive unit forms and their associated quasi-Cartan matrices, has lead to an Inflations Algorithm. However, this algorithm is not known to run in polynomial time. In this paper we use a so called flation transformation and show how its invariants can be used to characterize the Dynkin types A and D in the language of graph theory. Also, a polynomial-time algorithm for computing the Dynkin type is suggested.

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.