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

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 𝒰 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 𝔹 n , n ≥ 2 , ℂ n , n ≥ 3 , 𝔽 4 , 𝔾 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 ↦ 𝒟 Δ, with 𝒟 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 ⊂ 𝕄 n ( ℤ ) of matrix morsifications of the associated edge-bipartite Dynkin graph D = 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.