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Freeness of multi-reflection arrangements via primitive vector fields. (English) Zbl 1468.20075

Summary: In 2002, H. Terao [Invent. Math. 148, No. 3, 659–674 (2002; Zbl 1032.52013)] 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.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
14N20 Configurations and arrangements of linear subspaces
32S25 Complex surface and hypersurface singularities

Citations:

Zbl 1032.52013
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Full Text: DOI arXiv

References:

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