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Mesh Algorithms for Coxeter Spectral Classification of Cox-regular Edge-bipartite Graphs with Loops. [Part] 1, Mesh Root Systems

Kasjan S., Simson D.

Fundamenta Informaticae

2015

Vol. 139, nr 2

153--184

This is the first part of our two part paper with the same title. Following our Coxeter spectral study in [Fund. Inform. [123(2013), 447-490] and [SIAM J. Discr. Math. 27(2013), 827- 854] of the category UBigrn of loop-free edge-bipartite (signed) graphs Δ, with n ≥ 2 vertices, we study here the larger category RBigrn of Cox-regular edge-bipartite graphs Δ (possibly with dotted loops), up to the usual Z-congruences ~Z and ≈Z. The positive graphs Δ in RBigrn, with dotted loops, are studied by means of the complex Coxeter spectrum speccΔ C C, the irreducible mesh root systems of Dynkin types Bn, n ≥ 2, Cn, n ≥ 3, F4, G2, the isotropy group Gl(n;Z)Δ (containing the Weyl group of Δ), and by applying the matrix morsification technique introduced in [J. Pure A ppl. Algebra 215(2011), 13-24] and [Fund. Inform. [123(2013), 447-490]. One of our aims of the paper is to study the Coxeter spectral analysis question: "Does the congruence Δ ≈Z Δ' hold, for any pair of connected positive graphs Δ;Δ' ∊ RBigrn such that speccΔ = speccΔ' and the numbers of loops in Δ and Δ0 coincide?" We do it by a reduction to the Coxeter spectral study of the Gl(n, Z)D-orbits in the set MorD Mn(Z) of matrix morsifications of a Dynkin diagram D = DΔ ∊ UBigrn associated with Δ. In particular, we construct in the second part of the paper numeric algorithms for computing the connected positive edge-bipartite graphs Δ in RBigrn, for a fixed n ≥ 2, mesh algorithms for computing the set of all Z-invertible matrices B ∊ Gl(n;Z) definining the Z-congruence Δ ≈Z Δ', for positive graphs Δ;Δ' ∊ RBigrn, with n ≥ 2 fixed, and mesh-type algorithms for the mesh root systems Γ(RD Δ(RΔФΔ). In the first part of the paper we present an introduction to the study of Cox-regular edge-bipartite graphs Δ with dotted loops in relation with the irreducible reduced root systems and the Dynkin diagrams Bn, n ≥ 2, Cn, n ≥ 3, F4, G2. Moreover, we construct a unique ФD-mesh root system (RD,ФD) for each of the Cox-regular edge-bipartite graphs Bn, n ≥ 2, Cn, n ≥ 3, F4, calG2 of the type Bn, n ≥ 2, Cn, n ≥ 3, F4, G2, respectively. Our main inspiration for the study comes from the representation theory of posets, groups and algebras, Lie theory, and Diophantine geometry problems.

Mesh Algorithms for Coxeter Spectral Classification of Cox-regular Edge-bipartite Graphs with Loops [Part] 2. Application to Coxeter Spectral Analysis

Kasjan S., Simson D.

Fundamenta Informaticae

2015

Vol. 139, nr 2

185--209

This is a second part of our two part paper with the same title. Following our Coxeter spectral study in [Fund. Inform. [123(2013), 447-490] and [SIAM J. Discr. Math. 27(2013), 827- 854] of the category UBigrn of loop-free edge-bipartite (signed) graphs Δ, with n = 2 vertices, we study here the larger category RBigrn of Cox-regular edge-bipartite graphs Δ (possibly with dotted loops), up to the usual Z-congruences ~Z and Z. The positive graphs Δ in RBigrn, with dotted loops, are studied by means of the complex Coxeter spectrum speccΔ C, the irreducible mesh root systems of Dynkin types Bn, n = 2, Cn, n = 3, F4, G2, the isotropy group Gl(n, Z)Δ (containing the Weyl group of Δ), and by applying the matrix morsification technique introduced in [J. Pure Appl. Algebra 215(2011), 13-24] and [Fund. Inform. [123(2013), 447-490]. One of our aims of our two part paper is to study the Coxeter spectral analysis question: "Does the congruence Δ Z Δ' hold, for any pair of connected positive graphs Δ,Δ' ∊ RBigrn such that speccΔ = speccΔ' and the numbers of loops in ΔandΔ' coincide?"We do it by a reduction to the Coxeter spectral study of the Gl(n, Z)D-orbits in the set MorD C Mn(Z) of matrix morsifications of a Dynkin diagram D = DΔ ∊ UBigrn associated with Δ. In this second part, we construct numeric algorithms for computing the connected positive edge-bipartite graphs Δ in RBigrn, for a fixed n = 2, mesh algorithms for computing the set of all Z-invertible matrices B ∊ Gl(n, Z) definining the Z-congruenceΔ Z Δ', for positive graphsΔ,Δ' ∊ RBigrn, with n geq2 fixed, and mesh-type algorithms for the mesh root systems Γ(R·Δ,ΦΔ). We also present a classification and a structure type results for positive Cox-regular edge-bipartite graphs Δ with dotted loops.