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A Framework for Coxeter Spectral Analysis of Edge-bipartite Graphs, their Rational Morsifications and Mesh Geometries of Root Orbits

Simson D.

Fundamenta Informaticae

2013

Vol. 124, nr 3

309--338

Following the spectral Coxeter analysis of matrix morsifications for Dynkin diagrams, the spectral graph theory, a graph coloring technique, and algebraic methods in graph theory, we continue our study of the category UBigrn of loop-free edge-bipartite (signed) graphs ∆, with n > 2 vertices, by means of the Coxeter number oa, the Coxeter spectrum specc∆ of ∆, that is, the spectrum of the Coxeter polynomial cox∆(t) ∈ Z[t] and the Z-bilinear Gram form b∆ : Zn x Zn →Z of ∆ [SIAM J. Discrete Math. 27(2013)]. Our main inspiration for the study comes from the representation theory of posets, groups and algebras, Lie theory, and Diophantine geometry problems. We show that the Coxeter spectral classification of connected edge-bipartite graphs A in UBigrn reduces to the Coxeter spectral classification of rational matrix morsifications A ∈ MorD∆ for a simply-laced Dynkin diagram D∆ associated with ∆. Given ∆ in UBigrn, we study the isotropy subgroup Gl(n, Z)∆ of Gl(n, Z) that contains the Weyl group W∆. and acts on the set Mor∆ of rational matrix morsifications A of ∆ in such a way that the map A → (speccA, det A, c∆) is Gl(n, Z)∆-invariant. It is shown that, for n < 6, specc∆ is the spectrum of one of the Coxeter polynomials listed in Tables 3.11-3.11(a) (we determine them by computer search using symbolic and numeric computation). The question, if two connected positive edge-bipartite graphs ∆, ∆' in UBigrn, with specc∆= specc∆,, are Z-bilinear equivalent, is studied in the paper. The problem if any Z-invertible matrix A ∈ Mn(Z) is Z-congruent with its transpose Atr is also discussed.

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.