The freeness of ideal subarrangements of Weyl arrangements. (English) Zbl 1350.32028

Abe, Takuro; Barakat, Mohamed; Cuntz, Michael; Hoge, Torsten; Terao, Hiroaki

J. Eur. Math. Soc. (JEMS) 18, No. 6, 1339-1348 (2016).

Authors’ 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. 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. Our proof of the main theorem heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula.

Reviewer: Saeid Azam (Isfahan)

MSC:

52C35	Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
05E15	Combinatorial aspects of groups and algebras (MSC2010)
14N20	Configurations and arrangements of linear subspaces
17B22	Root systems
32S22	Relations with arrangements of hyperplanes

Keywords:

arrangements of hyperplanes; root system; Weyl arrangements; free arrangements

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