A pattern avoidance criterion for free inversion arrangements

TL;DR

This paper establishes a pattern avoidance criterion for the freeness of inversion arrangements in finite root systems, linking combinatorial pattern avoidance with algebraic properties of hyperplane arrangements.

Contribution

It provides a new characterization of free inversion arrangements via pattern avoidance and extends the Kostant-Shapiro-Steinberg rule to all types.

Findings

Freeness of hyperplane arrangements is characterized by pattern avoidance.

Freeness in root systems without type C or F factors is preserved under Peterson translation.

A generalized Kostant-Shapiro-Steinberg rule for coexponents is established.

Abstract

We show that the hyperplane arrangement of a coconvex set in a finite root system is free if and only if it is free in corank 4. As a consequence, we show that the inversion arrangement of a Weyl group element w is free if and only if w avoids a finite list of root system patterns. As a key part of the proof, we use a recent theorem of Abe and Yoshinaga to show that if the root system does not contain any factors of type C or F, then Peterson translation of coconvex sets preserves freeness. This also allows us to give a Kostant-Shapiro-Steinberg rule for the coexponents of a free inversion arrangement in any type.