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Discrete Mathematics & Theoretical Computer Science |
Takuro Abe ; Mohamed Barakat ; Michael Cuntz ; Torsten Hoge ; Hiroaki Terao - The freeness of ideal subarrangements of Weyl arrangements
dmtcs:2418 - Discrete Mathematics & Theoretical Computer Science, January 1, 2014, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) - https://doi.org/10.46298/dmtcs.2418
2; Michael Cuntz 3; Torsten Hoge 4; Hiroaki Terao 5
- 1 Department of Mechanical Engineering and Science, Kyoto University
- 2 Fachbereich Mathematik [Kaiserslautern]
- 3 Fakultät fur Mathematik und Physik [Hannover]
- 4 Fakultät für Mathematik [Bochum]
- 5 Department of Mathematics [Sapporo]
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.