TY - JOUR

T1 - Accurate arrangements

AU - Mücksch, Paul

AU - Röhrle, Gerhard

PY - 2021/6/4

Y1 - 2021/6/4

N2 - Let A be a Coxeter arrangement of rank ℓ. In 1987 Orlik, Solomon and Terao conjectured that for every 1≤d≤ℓ, the first d exponents of A – when listed in increasing order – are realized as the exponents of a free restriction of A to some intersection of reflecting hyperplanes of A of dimension d. This conjecture does follow from rather extensive case-by-case studies by Orlik and Terao from 1992 and 1993, where they show that all restrictions of Coxeter arrangements are free. We call a general free arrangement with this natural property involving their free restrictions accurate. In this paper we initialize their systematic study. Our principal result shows that MAT-free arrangements, a notion recently introduced by Cuntz and Mücksch, are accurate. This theorem in turn directly implies this special property for all ideal subarrangements of Weyl arrangements. In particular, this gives a new, simpler and uniform proof of the aforementioned conjecture of Orlik, Solomon and Terao for Weyl arrangements which is free of any case-by-case considerations. Another application of a slightly more general formulation of our main theorem shows that extended Catalan arrangements, extended Shi arrangements, and ideal-Shi arrangements share this property as well. We also study arrangements that satisfy a slightly weaker condition, called almost accurate arrangements, where we simply disregard the ordering of the exponents involved. This property in turn is implied by many well established concepts of freeness such as supersolvability and divisional freeness.

AB - Let A be a Coxeter arrangement of rank ℓ. In 1987 Orlik, Solomon and Terao conjectured that for every 1≤d≤ℓ, the first d exponents of A – when listed in increasing order – are realized as the exponents of a free restriction of A to some intersection of reflecting hyperplanes of A of dimension d. This conjecture does follow from rather extensive case-by-case studies by Orlik and Terao from 1992 and 1993, where they show that all restrictions of Coxeter arrangements are free. We call a general free arrangement with this natural property involving their free restrictions accurate. In this paper we initialize their systematic study. Our principal result shows that MAT-free arrangements, a notion recently introduced by Cuntz and Mücksch, are accurate. This theorem in turn directly implies this special property for all ideal subarrangements of Weyl arrangements. In particular, this gives a new, simpler and uniform proof of the aforementioned conjecture of Orlik, Solomon and Terao for Weyl arrangements which is free of any case-by-case considerations. Another application of a slightly more general formulation of our main theorem shows that extended Catalan arrangements, extended Shi arrangements, and ideal-Shi arrangements share this property as well. We also study arrangements that satisfy a slightly weaker condition, called almost accurate arrangements, where we simply disregard the ordering of the exponents involved. This property in turn is implied by many well established concepts of freeness such as supersolvability and divisional freeness.

KW - Coxeter arrangements

KW - Extended Shi arrangements

KW - Free arrangements

KW - Ideal arrangements

KW - MAT-free arrangements

KW - Weyl arrangements

UR - http://www.scopus.com/inward/record.url?scp=85103339540&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2021.107702

DO - 10.1016/j.aim.2021.107702

M3 - Article

VL - 383

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 383

M1 - 107702

ER -