# The freeness of ideal subarrangements of Weyl arrangements

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

Journal of the European Mathematical Society (2016)

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

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topAbe, 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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