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A Graph Theoretical Framework for the Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸 n

González Jiménez, Arturo Jesús

Fundamenta Informaticae

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2021

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Vol. 184, nr 1

49--82

EN

In the context of signed line graphs, this article introduces a modified inflation technique to study strong Gram congruence of non-negative (integral quadratic) unit forms, and uses it to show that weak and strong Gram congruence coincide among positive unit forms of Dynkin type An . The concept of inverse of a quiver is also introduced, and is used to obtain and analyze the Coxeter matrix of non-negative unit forms of Dynkin type An . With these tools, connected principal unit forms of Dynkin type An are also classified up to strong congruence.

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

Simson D.

Fundamenta Informaticae

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2013

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Vol. 123, nr 4

447--490

EN

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.

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Mesh Algorithms for Solving Principal Diophantine Equations, Sand-glass Tubes and Tori of Roots

Simson D.

Fundamenta Informaticae

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2011

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Vol. 109, nr 4

425-462

EN

We study integral solutions of diophantine equations q(x) = d, where x = (x1, . . . , xn), n ≥1, d .∈Z is an integer and q : Z^n →Z is a non-negative homogeneous quadratic form. Contrary to the negative solution of the Hilbert’s tenth problem, for any such a form q(x), we give efficient algorithms describing the set Rq(d) of all integral solutions of the equation q(x) = d in a Φ_A-mesh translation quiver form. We show in Section 5 that usually the set Rq(d) has a shape of a Φ_A-mesh sand-glass tube or of a A-mesh torus, see 5.8, 5.10, and 5.13. If, in addition, the subgroup Ker q = {v ∈Z^n; q(v) = 0} of Zn is infinite cyclic, we study the solutions of the equations q(x) = 1 by applying a defect δ_A : Z^n → Z and a reduced Coxeter number čA ∈ N defined by means of a morsification b_A : Zn × Zn → Z of q, see Section 4. On this way we get a simple graphical algorithm that constructs all integral solutions in the shape of a mesh translation oriented graph consisting of Coxeter A-orbits. It turns out that usually the graph has at most three infinite connected components and each of them has an infinite band shape, or an infinite horizontal tube shape, or has a sand-glass tube shape. The results have important applications in representation theory of groups, algebras, quivers and partially ordered sets, as well as in the study of derived categories (in the sense of Verdier) of module categories and categories of coherent sheaves over algebraic varieties.