Journal of the European Mathematical Society

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Volume 18, Issue 6, 2016, pp. 1339–1348
DOI: 10.4171/JEMS/615

Published online: 2016-04-25

The freeness of ideal subarrangements of Weyl arrangements

Takuro Abe[1], Mohamed Barakat[2], Michael Cuntz[3], Torsten Hoge[4] and Hiroaki Terao[5]

(1) Kyushu University, Fukuoka, Japan
(2) Universität Siegen, Germany
(3) Leibniz Universität Hannover, Germany
(4) Leibniz Universität Hannover, Germany
(5) Hokkaido University, Sapporo, Japan

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

Keywords: Arrangement of hyperplanes, root system,Weyl arrangement, free arrangement, ideals, dual partition theorem

Abe Takuro, Barakat Mohamed, Cuntz Michael, Hoge Torsten, Terao Hiroaki: The freeness of ideal subarrangements of Weyl arrangements. J. Eur. Math. Soc. 18 (2016), 1339-1348. doi: 10.4171/JEMS/615