The freeness of ideal subarrangements of Weyl arrangements

By Takuro Abe, Mohamed Barakat, Michael Cuntz, Torsten Hoge, Hiroaki Terao. 2013

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AbstractA Weyl arrangement is the arrangement defined by the root sys-tem of a finite Weyl group. When a set of positive roots is an ideal inthe root poset, we call the corresponding arrangement an ideal sub-arrangement. Our main theorem asserts that any ideal subarrange-ment is a free arrangement and that its exponents are given by thedual partition of the height distribution, which was conjectured bySommers-Tymoczko. In particular, when an ideal subarrangement isequal to the entire Weyl arrangement, our main theorem yields thecelebrated formula by Shapiro, Steinberg, Kostant, and Macdonald.Our proof of the main theorem heavily depends on the theory of freearrangements and thus greatly differs from the earlier proofs of theformula. 1 Introduction Let Φ be an irreducible root system of rank l and fix a simple system (orbasis) ∆ = {α 1 ,...,α l }. Let Φ + be the set of positive roots. Define thepartial order ≥ on Φ + such that α ≥ β if α − β ∈ Z ≥0 α 1 + ···+ Z ≥0 α l forα,β ∈ Φ + . A subset I of Φ

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

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