Supersolvability and the Koszul property of root ideal arrangements

TL;DR

This paper characterizes when root ideal arrangements are supersolvable, linking this property to the Koszul property of their Orlik-Solomon algebra, and identifies minimal non-supersolvable ideals.

Contribution

It provides a complete characterization of supersolvable root ideal arrangements and establishes the equivalence with the Koszul property of their Orlik-Solomon algebra.

Findings

Supersolvability corresponds to chain peelability of the ideal.

Minimal non-supersolvable ideals are identified in types D4 and F4.

Koszul property holds if and only if the arrangement is supersolvable.

Abstract

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 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.