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Toroidal Algorithms for Mesh Geometries of Root Orbits of the Dynkin Diagram D4

Simson D.

Fundamenta Informaticae

2013

Vol. 124, nr 3

339--364

By applying symbolic and numerical computation and the spectral Coxeter analysis technique of matrix morsifications introduced in our previous paper [Fund. Inform. 124(2013)], we present a complete algorithmic classification of the rational morsifications and their mesh geometries of root orbits for the Dynkin diagram 4 The structure of the isotropy group Gl(4, {Z})D4 of D 4 is also studied. As a byproduct of our technique we show that, given a connected loop-free positive edge-bipartite graph Δ, with n ≥ 4 vertices (in the sense of our paper [SIAM J. Discrete Math. 27(2013)]) and the positive definite Gram unit formqΔ ; Zn→Z, any positive integer d ≥ 1 can be presented as d = qΔ(v), with v Є Zn In case n = 3, a positive integer d ≥ 1 can be presented as d = qΔ(v), with v Є Zn , if and only if d is not of the form 4a(16 · b + 14), where a and b are non-negative integers.

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.