Arrangements of ideal type

TL;DR

This paper investigates the freeness and Poincaré polynomial properties of arrangements of ideal type in root systems, establishing inductive freeness broadly and confirming a conjectured polynomial factorization in most cases.

Contribution

It proves that all arrangements of ideal type are inductively free except possibly in some E_8 cases and confirms the multiplicative Poincaré polynomial formula in most instances.

Findings

All arrangements of ideal type are inductively free, with potential exceptions in type E_8.

The Poincaré polynomial I(t) satisfies a multiplicative formula in almost all cases.

The dual of the height partition determines the exponents of the free arrangements.

Abstract

In 2006 Sommers and Tymoczko defined so called arrangements of ideal type A_I stemming from ideals I in the set of positive roots of a reduced root system. They showed in a case by case argument that A_I is free if the root system is of classical type or G_2 and conjectured that this is also the case for all types. This was established only recently in a uniform manner by Abe, Barakat, Cuntz, Hoge and Terao. The set of non-zero exponents of the free arrangement A_I is given by the dual of the height partition of the roots in the complement of I in the set of positive roots, generalizing the Shapiro-Steinberg-Kostant theorem. Our first aim in this paper is to investigate a stronger freeness property of the A_I. We show that all A_I are inductively free, with the possible exception of some cases in type E_8. In the same paper, Sommers and Tymoczko define a Poincar\'e polynomial I(t) associated with each ideal I which generalizes the Poincar\'e polynomial W(t) for the underlying Weyl group W. Solomon showed that W(t) satisfies a product decomposition depending on the exponents of W for any Coxeter group W. Sommers and Tymoczko showed in a case by case analysis in type A, B and C, and some small rank exceptional types that a similar factorization property holds for the Poincar\'e polynomials I(t) generalizing the formula of Solomon for W(t). They conjectured that their multiplicative formula for I(t) holds in all types. Here we show that this conjecture holds inductively in almost all instances.