Supersolvability and the Koszul property of root ideal arrangements
2016 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 144, 1401-1413 p.Article in journal (Refereed) Published
A root ideal arrangement A_I is the set of reflecting hyperplanes corresponding to the roots in an order ideal I of the root poset on the positive roots of a finite crystallographic root system. A characterisation of supersolvable root ideal arrangements is obtained. Namely, A_I is supersolvable if and only if I is chain peelable, meaning that it is possible to reach the empty poset from I by in each step removing a maximal chain which is also an order filter. In particular, supersolvability is preserved under taking subideals. We identify the maximal ideals that correspond to non-supersolvable arrangements. There are essentially two such ideals, one in type D_4 and one in type F_4. By showing that A_I is not line-closed if I contains one of these, we deduce that the Orlik-Solomon algebra OS(A_I) has the Koszul property if and only if A_I is supersolvable.
Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2016. Vol. 144, 1401-1413 p.
hyperplane arrangement, root poset, supersolvability, Koszul algebra
Discrete Mathematics Geometry
IdentifiersURN: urn:nbn:se:liu:diva-124535DOI: 10.1090/proc/12810ISI: 000369298400003OAI: oai:DiVA.org:liu-124535DiVA: diva2:899516