SUPERSOLVABILITY AND THE KOSZUL PROPERTY OF ROOT IDEAL ARRANGEMENTS

A root ideal arrangement AI is the set of reflecting hyperplanes corresponding to the roots in an order ideal I subset of Phi(+) of the root poset on the positive roots of a finite crystallographic root system Phi. 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 undertaking subideals. We identify the minimal 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.