TY - JOUR

T1 - The freeness of ideal subarrangements of Weyl arrangements

AU - Abe, Takuro

AU - Barakat, Mohamed

AU - Cuntz, Michael

AU - Hoge, Torsten

AU - Terao, Hiroaki

N1 - Funding Information:
The authors are grateful to Naoya Enomoto, Louis Solomon, Akimichi Takemura and Masahiko Yoshinaga for useful and stimulating discussions. They are indebted to Eric Sommers who kindly let them know about the Sommers-Tymoczko conjecture. They express their gratitude to the referee who suggested appropriate changes in their earlier version. Also, they thank Gerhard Röhrle for his hospitality during their stay in Bochum. The first author is supported by JSPS Grants-in-Aid for Young Scientists (B) No. 24740012. The last author is supported by JSPS Grants-in-Aid for basic research (A) No. 24244001.
Publisher Copyright:
© European Mathematical Society 2016.

PY - 2016

Y1 - 2016

N2 - 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.

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 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.

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U2 - 10.4171/JEMS/615

DO - 10.4171/JEMS/615

M3 - Article

AN - SCOPUS:85012069108

SN - 1435-9855

VL - 18

SP - 1339

EP - 1348

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 6

ER -