Algorithms for Isotropy Groups of Cox-regular Edge-bipartite Graphs

• Published in: Fundamenta informaticae Vol. 139; no. 3; pp. 249 - 275
• Main Authors: Kasjan, Stanisław, Simson, Daniel
• Format: Journal Article
• Language: English
• Published: London, England SAGE Publications 01.08.2015
• Subjects: Coxeter spectrum; Dynkin diagram; isotropy group; Coxeter-Dynkin type; poset
• ISSN: 0169-2968; 1875-8681
• DOI: 10.3233/FI-2015-1234

This paper can be viewed as a third part of our paper [Fund. Inform. 2015, in press]. Following our Coxeter spectral study in [Fund. Inform. 123(2013), 447-490] and [SIAM J. Discr. Math. 27(2013), 827-854] of the category ℬ i g r n of loop-free edge-bipartite (signed) graphs Δ, with n ≥ 2 vertices, we study a larger category ℛ ℬ i g r n 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 ℛ ℬ i g r n , with dotted loops, are studied by means of the complex Coxeter spectrum s p e c c Δ   ⊂ ℂ , the irreducible mesh root systems of Dynkin types n ,   n ≥ 2 , ℂ n ,   n ≥ 3 ,   4 ,   2 , 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] Here we present combinatorial algorithms for constructing the isotropy groups G1 ( n ,   ℤ ) Δ . One of the aims of our three paper series is to develop computational tools for the study of the ℤ-congruence ∼ℤ and the following Coxeter spectral analysis question: “Does the congruence Δ ≈ℤ Δ′ holds, for any pair of connected positive graphs Δ ,   Δ ′ ∈ ℛ ℬ i g r n such that spec c Δ = spec c Δ ′ and the numbers of loops in Δ and Δ′ coincide?” For this purpose, we construct in this paper a extended inflation algorithm Δ ↦ Δ , with Δ ∼ ℤ Δ , that allows a reduction of the question to the Coxeter spectral study of the G1 ( n ,   ℤ ) D -orbits in the set M o r D   ⊂   n ( ℤ ) of matrix morsifications of the associated edge-bipartite Dynkin graph D = Δ ∈ ℛ B i g r n . We also outline a construction of a numeric algorithm for computing the isotropy group G1 ( n ,   ℤ ) Δ of any connected positive edge-bipartite graph Δ in ℛ ℬ i g r n . Finally, we compute the finite isotropy group G1 ( n ,   ℤ ) D , for each of the Cox-regular edge-bipartite Dynkin graphs D.