The freeness of ideal subarrangements of Weyl arrangements

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  • Kyushu University
  • University of Siegen
  • Hokkaido University
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Original languageEnglish
Pages (from-to)1339-1348
Number of pages10
JournalJournal of the European Mathematical Society
Volume18
Issue number6
Publication statusPublished - 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

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Cite this

The freeness of ideal subarrangements of Weyl arrangements. / Abe, Takuro; Barakat, Mohamed; Cuntz, Michael et al.
In: Journal of the European Mathematical Society, Vol. 18, No. 6, 25.04.2016, p. 1339-1348.

Research output: Contribution to journalArticleResearchpeer review

Abe T, Barakat M, Cuntz M, Hoge T, Terao H. The freeness of ideal subarrangements of Weyl arrangements. Journal of the European Mathematical Society. 2016 Apr 25;18(6):1339-1348. doi: 10.4171/JEMS/615, 10.15488/2358
Abe, Takuro ; Barakat, Mohamed ; Cuntz, Michael et al. / The freeness of ideal subarrangements of Weyl arrangements. In: Journal of the European Mathematical Society. 2016 ; Vol. 18, No. 6. pp. 1339-1348.
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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.

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