# The freeness of ideal subarrangements of Weyl arrangements

• Volume: 018, Issue: 6, page 1339-1348
• ISSN: 1435-9855

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## Abstract

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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 entireWeyl 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.

## How to cite

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Abe, Takuro, et al. "The freeness of ideal subarrangements of Weyl arrangements." Journal of the European Mathematical Society 018.6 (2016): 1339-1348. <http://eudml.org/doc/277163>.

@article{Abe2016,
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 entireWeyl 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.},
author = {Abe, Takuro, Barakat, Mohamed, Cuntz, Michael, Hoge, Torsten, Terao, Hiroaki},
journal = {Journal of the European Mathematical Society},
keywords = {arrangement of hyperplanes; root system; Weyl arrangement; free arrangement; ideals; dual partition theorem; arrangement of hyperplanes; root system; Weyl arrangement; free arrangement; ideals; dual partition theorem},
language = {eng},
number = {6},
pages = {1339-1348},
publisher = {European Mathematical Society Publishing House},
title = {The freeness of ideal subarrangements of Weyl arrangements},
url = {http://eudml.org/doc/277163},
volume = {018},
year = {2016},
}

TY - JOUR
AU - Abe, Takuro
AU - Barakat, Mohamed
AU - Cuntz, Michael
AU - Hoge, Torsten
AU - Terao, Hiroaki
TI - The freeness of ideal subarrangements of Weyl arrangements
JO - Journal of the European Mathematical Society
PY - 2016
PB - European Mathematical Society Publishing House
VL - 018
IS - 6
SP - 1339
EP - 1348
AB - 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 entireWeyl 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.
LA - eng
KW - arrangement of hyperplanes; root system; Weyl arrangement; free arrangement; ideals; dual partition theorem; arrangement of hyperplanes; root system; Weyl arrangement; free arrangement; ideals; dual partition theorem
UR - http://eudml.org/doc/277163
ER -

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