The freeness of ideal subarrangements of Weyl arrangements
| Autor: | Torsten Hoge, Mohamed Barakat, Takuro Abe, Hiroaki Terao, Michael Cuntz |
|---|---|
| Přispěvatelé: | Department of Mechanical Engineering and Science, Kyoto University, Department of Mechanical Engineering and Science, Kyoto University [Kyoto]-Kyoto University [Kyoto], Fachbereich Mathematik [Kaiserslautern], Technische Universität Kaiserslautern (TU Kaiserslautern), Fakultät fur Mathematik und Physik [Hannover], Leibniz Universität Hannover [Hannover] (LUH), Fakultät für Mathematik [Bochum], Ruhr-Universität Bochum [Bochum], Department of Mathematics [Sapporo], Hokkaido University [Sapporo, Japan], Louis J. Billera and Isabella Novik |
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Pure mathematics
General Computer Science Root system General Mathematics ideal 0102 computer and information sciences [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] Mathematical proof 01 natural sciences free arrangements Theoretical Computer Science Combinatorics Free arrangement symbols.namesake [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] FOS: Mathematics Weyl arrangement Mathematics - Combinatorics Discrete Mathematics and Combinatorics Partition (number theory) Arrangement of hyperplanes exponents Representation Theory (math.RT) ddc:510 0101 mathematics Mathematics::Representation Theory 32S22 17B22 05A18 Mathematics Weyl group Applied Mathematics 010102 general mathematics Dewey Decimal Classification::500 | Naturwissenschaften::510 | Mathematik [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] Dual partition theorem [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM] 010201 computation theory & mathematics symbols Combinatorics (math.CO) Partially ordered set Mathematics - Representation Theory Ideals height |
| Zdroj: | Discrete Mathematics and Theoretical Computer Science 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.501-512 Journal of the European Mathematical Society 18 (2016), Nr. 6 |
| ISSN: | 1462-7264 1365-8050 |
| DOI: | 10.15488/2358 |
| Popis: | 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. Un arrangement de Weyl est défini par l’arrangement d’hyperplans du système de racines d’un groupe de Weyl fini. Quand un ensemble de racines positives est un idéal dans le poset de racines, nous appelons l’arrangement correspondant un sous-arrangement idéal. Notre théorème principal affirme que tout sous-arrangement idéal est un arrangement libre et que ses exposants sont donnés par la partition duale de la distribution des hauteurs, ce qui avait été conjecturé par Sommers-Tymoczko. En particulier, quand le sous-arrangement idéal est égal à l’arrangement de Weyl, notre théorème principal donne la célèbre formule par Shapiro, Steinberg, Kostant et Macdonald. La démonstration du théorème principal n’utilise pas de classification. Elle dépend fortement de la théorie des arrangements libres et diffère ainsi grandement des démonstrations précédentes de la formule. |
| Databáze: | OpenAIRE |
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