Browsing by Author "Stump, Christian"

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    Cataland: Why the Fuß?
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2019) Stump, Christian; Thomas, Hugh; Williams, Nathan
    The three main objects in noncrossing Catalan combinatorics associated to a finite Coxeter system are noncrossing partitions, clusters, and sortable elements. The first two of these have known Fuß-Catalan generalizations. We provide new viewpoints for both and introduce the missing generalization of sortable elements by lifting the theory from the Coxeter system to the associated positive Artin monoid. We show how this new perspective ties together all three generalizations, providing a uniform framework for noncrossing Fuß-Catalan combinatorics. Having developed the combinatorial theory, we provide an interpretation of our generalizations in the language of the representation theory of hereditary Artin algebras.
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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.