Free filtrations of affine Weyl arrangements and the ideal-Shi arrangements
Takuro Abe, Hiroaki Terao
TL;DR
This paper proves that ideal-Shi arrangements are free hyperplane arrangements satisfying the dual-partition formula, and establishes a saturated free filtration of affine Weyl arrangements, extending previous conjectures.
Contribution
It introduces a new free filtration of affine Weyl arrangements composed of free subarrangements satisfying the dual-partition formula.
Findings
Ideal-Shi arrangements are free central arrangements.
Existence of a saturated free filtration for affine Weyl arrangements.
Generalization of a conjecture by Sommers and Tymoczko.
Abstract
In this article we prove that the ideal-Shi arrangements are free central arrangements of hyperplanes satisfying the dual-partition formula. Then it immediately follows that there exists a saturated free filtration of the cone of any affine Weyl arrangement such that each filter is a free subarrangement satisfying the dual-partition formula. This generalizes the main result in \cite{ABCHT} which affirmatively settled a conjecture by Sommers and Tymoczko \cite{SomTym}.
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