Details
Original language | English |
---|---|
Pages (from-to) | 1339-1348 |
Number of pages | 10 |
Journal | Journal of the European Mathematical Society |
Volume | 18 |
Issue number | 6 |
Publication status | Published - 25 Apr 2016 |
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 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.
Keywords
- Arrangement of hyperplanes, Dual partition theorem, Free arrangement, Ideals, Root system, Weyl arrangement
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
- Mathematics(all)
- Applied Mathematics
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In: Journal of the European Mathematical Society, Vol. 18, No. 6, 25.04.2016, p. 1339-1348.
Research output: Contribution to journal › Article › Research › peer review
}
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.
PY - 2016/4/25
Y1 - 2016/4/25
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.
KW - Arrangement of hyperplanes
KW - Dual partition theorem
KW - Free arrangement
KW - Ideals
KW - Root system
KW - Weyl arrangement
UR - http://www.scopus.com/inward/record.url?scp=85012069108&partnerID=8YFLogxK
U2 - 10.4171/JEMS/615
DO - 10.4171/JEMS/615
M3 - Article
AN - SCOPUS:85012069108
VL - 18
SP - 1339
EP - 1348
JO - Journal of the European Mathematical Society
JF - Journal of the European Mathematical Society
SN - 1435-9855
IS - 6
ER -