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Author: Hoge, Torsten × Author: Röhrle, Gerhard × Type: Text × WGL Institute: MFO ×

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    Inductive freeness of Ziegler’s canonical multiderivations for reflection arrangements
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Hoge, Torsten; Röhrle, Gerhard
    Let A be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction A 00 of A to any hyperplane endowed with the natural multiplicity is then a free multiarrangement. We initiate a study of the stronger freeness property of inductive freeness for these canonical free multiarrangements and investigate them for the underlying class of re ection arrangements. More precisely, let A = A (W) be the re ection arrangement of a complex re ection group W. By work of Terao, each such re ection arrangement is free. Thus so is Ziegler's canonical multiplicity on the restriction A 00 of A to a hyperplane. We show that the latter is inductively free as a multiarrangement if and only if A 00 itself is inductively free.
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    Freeness of multi-reflection arrangements via primitive vector fields
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Hoge, Torsten; Mano, Toshiyuki; Röhrle, Gerhard; Stump, Christian
    In 2002, Terao showed that every reection multi-arrangement of a real reection group with constant multiplicity is free by providing a basis of the module of derivations. We rst generalize Terao's result to multi-arrangements stemming from well-generated unitary reection groups, where the multiplicity of a hyperplane depends on the order of its stabilizer. Here the exponents depend on the exponents of the dual reection representation. We then extend our results further to all imprimitive irreducible unitary reection groups. In this case the exponents turn out to depend on the exponents of a certain Galois twist of the dual reection representation that comes from a Beynon-Lusztig type semi-palindromicity of the fake degrees.