The freeness of ideal subarrangements of Weyl arrangements

Takuro Abe,Michael Cuntz,Hiroaki Terao,Mohamed Barakat,Torsten Hoge

doi:10.46298/dmtcs.2418

Abstract

A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. The proof of the main theorem is classification-free. It heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula. Un arrangement de Weyl est défini par l’arrangement d’hyperplans du système de racines d’un groupe de Weyl fini. Quand un ensemble de racines positives est un idéal dans le poset de racines, nous appelons l’arrangement correspondant un sous-arrangement idéal. Notre théorème principal affirme que tout sous-arrangement idéal est un arrangement libre et que ses exposants sont donnés par la partition duale de la distribution des hauteurs, ce qui avait été conjecturé par Sommers-Tymoczko. En particulier, quand le sous-arrangement idéal est égal à l’arrangement de Weyl, notre théorème principal donne la célèbre formule par Shapiro, Steinberg, Kostant et Macdonald. La démonstration du théorème principal n’utilise pas de classification. Elle dépend fortement de la théorie des arrangements libres et diffère ainsi grandement des démonstrations précédentes de la formule.

Highlights

Let Φ be an irreducible root system of rank and fix a simple system ∆ = {α1, . . . , α }
A subset I of Φ+ is called an ideal if each positive root β satisfying α ≥ β for some α ∈ I belongs to I
The height distribution in I is a sequence of positive integers (i1, i2, . . . , im), where ij := |{α ∈ I | ht(α) = j}|

Summary

Introduction

The height ht(α) of a positive root α = i=1 ciαi is defined to be i=1 ci. Theorem 1.1 was conjectured by Sommers and Tymoczko in [12] where they defined and studied the ideal exponents, which is essentially the same as our DP(I). If we set I = Φ+ in Theorem 1.1, we get the following: Corollary 1.2 (Steinberg [13], Kostant [6], Macdonald [7]) The exponents of the Weyl arrangement A(Φ+) are given by DP(Φ+). Φ+j is an ideal (which we call the j-th height ideal) and the arrangement A(Φ+j ) is free with the exponents DP(Φ+j ). Corollary 1.7 For any ideal I ⊆ Φ+, let A(I)C denote the complexified arrangement of A(I).

Preliminaries

Multiple addition theorem