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

Abstract

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 $𝒰ℬig{r}_n$ of loop-free edge-bipartite (signed) graphs Δ, with n ≥ 2 vertices, we study a larger category $ℛℬig{r}_n$ 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 $ℛℬig{r}_n$, with dotted loops, are studied by means of the complex Coxeter spectrum $spec{c}_{\Delta }\subset ℂ$, the irreducible mesh root systems of Dynkin types ${𝔹}_n,n\ge 2,{ℂ}_n,n\ge 3,{𝔽}_{\text{4}},{𝔾}_{\text{2}}$, 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 $\text{G1}(n,\mathbb{Z}{)}_{\Delta }$.

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 $\Delta ,{\Delta }^{\prime }\in ℛℬig{r}_n$ such that $\text{spec}{\text{c}}_{\Delta }=\text{spec}{\text{c}}_{{\Delta }^{\prime }}$ and the numbers of loops in Δ and Δ′ coincide?”

For this purpose, we construct in this paper a extended inflation algorithm $\Delta \mapsto 𝒟\Delta$, with $𝒟\Delta {\sim }_{\mathbb{Z}}\Delta$, that allows a reduction of the question to the Coxeter spectral study of the $\text{G1}(n,\mathbb{Z}{)}_D$-orbits in the set $Mo{r}_D\subset {𝕄}_n(\mathbb{Z})$ of matrix morsifications of the associated edge-bipartite Dynkin graph $D=𝒟\Delta \in ℛBig{r}_n$. We also outline a construction of a numeric algorithm for computing the isotropy group $\text{G1}(n,\mathbb{Z}{)}_{\Delta }$ of any connected positive edge-bipartite graph Δ in $ℛℬig{r}_n$. Finally, we compute the finite isotropy group $\text{G1}(n,\mathbb{Z}{)}_D$, for each of the Cox-regular edge-bipartite Dynkin graphs D.