Affiliations: [a] Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland. {skasjan,simson}@mat.umk.pl
Note: [*] Supported by Polish Research Grant NCN 2011/03/B/ST1/00824.
Note: [1] Address for correspondence: Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
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 𝒰ℬigrn of loop-free edge-bipartite (signed) graphs Δ, with n ≥ 2 vertices, we study a 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 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′∈ℛℬigrn such that speccΔ=speccΔ′ and the numbers of loops in Δ and Δ′ coincide?” For this purpose, we construct in this paper a extended inflation algorithm Δ↦𝒟Δ, with 𝒟Δ∼ℤΔ, that allows a reduction of the question to the Coxeter spectral study of the G1(n, ℤ)D-orbits in the set MorD ⊂ 𝕄n(ℤ) of matrix morsifications of the associated edge-bipartite Dynkin graph D=𝒟Δ∈ℛBigrn. 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 ℛℬigrn. Finally, we compute the finite isotropy group G1(n, ℤ)D, for each of the Cox-regular edge-bipartite Dynkin graphs D.
Keywords: poset, Coxeter spectrum, Dynkin diagram, Coxeter-Dynkin type, isotropy group
