Projective dimension of weakly chordal graphic arrangements

authored by

Takuro Abe, Lukas Kühne, Paul Mücksch, Leonie Mühlherr

Abstract

A graphic arrangement is a subarrangement of the braid arrangement whose set of hyperplanes is determined by an undirected graph. A classical result due to Stanley, Edelman and Reiner states that a graphic arrangement is free if and only if the corresponding graph is chordal, i.e., the graph has no chordless cycle with four or more vertices. In this article we extend this result by proving that the module of logarithmic derivations of a graphic arrangement has projective dimension at most one if and only if the corresponding graph is weakly chordal, i.e., the graph and its complement have no chordless cycle with five or more vertices.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

Rikkyo University
Bielefeld University

Type

Article

Journal

Seminaire Lotharingien de Combinatoire

Volume

8

Publication date

03.03.2025

Publication status

E-pub ahead of print

Peer reviewed

Yes

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

Electronic version(s)

https://doi.org/10.5802/alco.403 (Access: Open)
https://doi.org/10.48550/arXiv.2307.06021 (Access: Open)