Freeness of multi-reflection arrangements via primitive vector fields
| Autor: | Torsten Hoge, Toshiyuki Mano, Christian Stump, Gerhard Röhrle |
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| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
General Mathematics 010102 general mathematics Multiplicity (mathematics) Group Theory (math.GR) Mathematics - Commutative Algebra Commutative Algebra (math.AC) 01 natural sciences Unitary state 20F55 52C35 14N20 32S25 32S22 05E15 Hyperplane 0103 physical sciences FOS: Mathematics Mathematics - Combinatorics Vector field Combinatorics (math.CO) 010307 mathematical physics 0101 mathematics Twist Reflection group Mathematics - Group Theory Mathematics |
| Zdroj: | Advances in Mathematics. 350:63-96 |
| ISSN: | 0001-8708 |
| DOI: | 10.1016/j.aim.2019.04.044 |
| Popis: | In 2002, Terao showed that every reflection multi-arrangement of a real reflection group with constant multiplicity is free by providing a basis of the module of derivations. We first generalize Terao's result to multi-arrangements stemming from well-generated unitary reflection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reflection representation. We then extend our results further to all imprimitive irreducible unitary reflection groups. In this case the exponents turn out to depend on the exponents of a certain Galois twist of the dual reflection representation that comes from a Beynon-Lusztig type semi-palindromicity of the fake degrees. Comment: 26 pages; v2: an analogue of Saito's Hodge filtration is introduced and is used to show the universality of the vector field \nabla_D^{-m}(E) in Prop. 3.18; v3: minor changes; v4: reference [KMS18b] added; other minor changes; final version to appear in Advances in Math |
| Databáze: | OpenAIRE |
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