TY - JOUR

T1 - A bound for crystallographic arrangements

AU - Cuntz, Michael

PY - 2021/5/15

Y1 - 2021/5/15

N2 - A crystallographic arrangement is a set of linear hyperplanes satisfying a certain integrality property and decomposing the space into simplicial cones. Crystallographic arrangements were completely classified in a series of papers by Heckenberger and the author. However, this classification is based on two computer proofs checking millions of cases. In the present paper, we prove without using a computer that, up to equivalence, there are only finitely many irreducible crystallographic arrangements in each rank greater than two.

AB - A crystallographic arrangement is a set of linear hyperplanes satisfying a certain integrality property and decomposing the space into simplicial cones. Crystallographic arrangements were completely classified in a series of papers by Heckenberger and the author. However, this classification is based on two computer proofs checking millions of cases. In the present paper, we prove without using a computer that, up to equivalence, there are only finitely many irreducible crystallographic arrangements in each rank greater than two.

KW - Reflection group

KW - Simplicial arrangement

KW - Weyl group

KW - Weyl groupoid

UR - http://www.scopus.com/inward/record.url?scp=85100250719&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2021.01.028

DO - 10.1016/j.jalgebra.2021.01.028

M3 - Article

VL - 574

SP - 50

EP - 69

JO - Journal of algebra

JF - Journal of algebra

SN - 0021-8693

ER -