Vertex-weighted digraphs and freeness of arrangements between Shi and Ish

authored by: Takuro Abe, Tan Nhat Tran, Shuhei Tsujie
Abstract: We introduce and study a digraph analogue of Stanley's ψ-graphical arrangements from the perspectives of combinatorics and freeness. Our arrangements form a common generalization of various classes of arrangements in the literature including the Catalan arrangement, the Shi arrangement, the Ish arrangement, and especially the arrangements interpolating between Shi and Ish, recently introduced by Duarte and Guedes de Oliveira. The arrangements between Shi and Ish all are proved to have the same characteristic polynomial with all integer roots, thus raising the natural question of their freeness. We define two operations on digraphs, which we shall call king and coking elimination operations and prove that subject to certain conditions on the weight ψ, the operations preserve the characteristic polynomials and freeness of the associated arrangements. As an application, we affirmatively prove that the arrangements between Shi and Ish all are free, and among them only the Ish arrangement has supersolvable cone.
Organisation(s): Institute of Algebra, Number Theory and Discrete Mathematics
External Organisation(s): Rikkyo University
Hokkaido University of Education
Type: Article
Journal: European Journal of Combinatorics
Volume: 118
ISSN: 0195-6698
Publication date: 05.2024
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Discrete Mathematics and Combinatorics
Electronic version(s): https://doi.org/10.48550/arXiv.2108.02518 (Access: Open)
https://doi.org/10.1016/j.ejc.2024.103920 (Access: Closed)

BibTeX

@article{498403f7271344ce9b661314c222f84b,
title = "Vertex-weighted digraphs and freeness of arrangements between Shi and Ish",
abstract = "We introduce and study a digraph analogue of Stanley's ψ-graphical arrangements from the perspectives of combinatorics and freeness. Our arrangements form a common generalization of various classes of arrangements in the literature including the Catalan arrangement, the Shi arrangement, the Ish arrangement, and especially the arrangements interpolating between Shi and Ish, recently introduced by Duarte and Guedes de Oliveira. The arrangements between Shi and Ish all are proved to have the same characteristic polynomial with all integer roots, thus raising the natural question of their freeness. We define two operations on digraphs, which we shall call king and coking elimination operations and prove that subject to certain conditions on the weight ψ, the operations preserve the characteristic polynomials and freeness of the associated arrangements. As an application, we affirmatively prove that the arrangements between Shi and Ish all are free, and among them only the Ish arrangement has supersolvable cone.",
author = "Takuro Abe and Tran, {Tan Nhat} and Shuhei Tsujie",
note = "Funding Information: We would like to thank Professor Masahiko Yoshinaga for suggesting to us the question of freeness for the arrangements between Shi and Ish. The first author is partially supported by JSPS KAKENHI grant numbers JP20K20880 and 21H00975 . The second author was supported by JSPS Research Fellowship for Young Scientists grant number 19J12024 at Hokkaido University and a postdoctoral fellowship of the Alexander von Humboldt Foundation at Ruhr-Universit{\"a}t Bochum . ",
year = "2024",
month = may,
doi = "10.48550/arXiv.2108.02518",
language = "English",
volume = "118",
journal = "European Journal of Combinatorics",
issn = "0195-6698",
publisher = "Academic Press Inc.",
}