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Coxeter Invariants for Non-negative Unit Forms of Dynkin Type A

Jiménez González Jesús Arturo

Fundamenta Informaticae

2022

Vol. 185, nr 3

221--246

Two integral quadratic unit forms are called strongly Gram congruent if their upper triangular Gram matrices are ℤ-congruent. The paper gives a combinatorial strong Gram invariant for those unit forms that are non-negative of Dynkin type r (for r ≥ 1), within the framework introduced in [Fundamenta Informaticae 184(1):49–82, 2021], and uses it to determine all corresponding Coxeter polynomials and (reduced) Coxeter numbers.

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

2021

Vol. 184, nr 1

49--82

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.

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

Simson D.

Fundamenta Informaticae

2011

Vol. 109, nr 4

425-462

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.