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

15.01.2024

Publication status

E-pub ahead of print

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)