Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphs

Tan Nhat Tran,Akiyoshi Tsuchiya

doi:10.5802/crmath.210

Abstract

The class of Worpitzky-compatible subarrangements of a Weyl arrangement together with an associated Eulerian polynomial was recently introduced by Ashraf, Yoshinaga and the first author, which brings the characteristic and Ehrhart quasi-polynomials into one formula. The subarrangements of the braid arrangement, the Weyl arrangement of type A, are known as the graphic arrangements. We prove that the Worpitzky-compatible graphic arrangements are characterized by cocomparability graphs. This can be regarded as a counterpart of the characterization by Stanley and Edelman–Reiner of free and supersolvable graphic arrangements in terms of chordal graphs. Our main result yields new formulas for the chromatic and graphic Eulerian polynomials of cocomparability graphs.

Highlights

Let V be an -dimensional Euclidean vector space with the standard inner product (·, ·)
For Ψ ⊆ Φ+, the Weyl subarrangement AΨ is defined by AΨ := {Hα,0 | α ∈ Ψ}
Characterizing free subarrangements of an arbitrary Weyl arrangement is still a challenging problem, in the case of braid arrangement, the free subarrangements can be completely analyzed using the connection to graphs. It follows from the works of Stanley [21] and Edelman– Reiner [6] that free and supersolvable graphic arrangements are synonyms, and they correspond to chordal graphs

Summary

Introduction

Let V be an -dimensional Euclidean vector space with the standard inner product (·, ·). Various free subarrangements of a Weyl arrangement of type B were studied, e.g., [6, 13, 23]. Characterizing free subarrangements of an arbitrary Weyl arrangement is still a challenging problem, in the case of braid arrangement, the free subarrangements can be completely analyzed using the connection to graphs It follows from the works of Stanley [21] and Edelman– Reiner [6] that free and supersolvable graphic arrangements are synonyms, and they correspond to chordal graphs (every induced cycle in the graph has exactly 3 vertices). Our main result connects the geometric property of a graphic arrangement with the combinatorial property of the underlying graph. Our main result yields a correspondence between interval graphs and graphic arrangements that are compatible and free. Location cocomparability compatible (= strongly compatible) Theorems 3, 9 chordal interval unit interval free (= supersolvable) compatible ∩ free ideal [6, 21] Corollary 15 Theorem 16

Proof of the main result

Application to graph polynomials of Cocomparability graphs