The freeness of ideal subarrangements of Weyl arrangements
Takuro Abe
Kyushu University, Fukuoka, JapanMohamed Barakat
Universität Siegen, GermanyMichael Cuntz
Leibniz Universität Hannover, GermanyTorsten Hoge
Leibniz Universität Hannover, GermanyHiroaki Terao
Hokkaido University, Sapporo, Japan
![The freeness of ideal subarrangements of Weyl arrangements cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-jems-volume-18-issue-6.png&w=3840&q=90)
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
Cite this article
Takuro Abe, Mohamed Barakat, Michael Cuntz, Torsten Hoge, Hiroaki Terao, The freeness of ideal subarrangements of Weyl arrangements. J. Eur. Math. Soc. 18 (2016), no. 6, pp. 1339–1348
DOI 10.4171/JEMS/615