Browsing by Author "Zieschang, Paul-Hermann"

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    Hypergroups and Twin Buildings, I
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) French, Christopher; Zieschang, Paul-Hermann
    We discuss a conjecture on thick twin buildings the verification of which is needed in order to show that thick twin buildings are mathematically equivalent to regular actions of certain twin Coxeter hypergroups. (A corresponding result for buildings is shown in [5; Sections 10.2, 10.3].) We prove that the conjecture holds in the case where the support of its sagittal has cardinality 2 and in the case where its sagittal has length at most 3. (Sagittals are defined in Section 1.) Our exposition is based on an earlier treatment of the subject; cf. [3].
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    Representation theory of imprimitive of non-commutative association schemes of order
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Hanaki, Akihide; Zieschang, Paul-Hermann
    In this note, we investigate association schemes of order 6. We prove that non-normal closed subsets of such schemes have order 2 and that normal closed subsets of non-commutative schemes have order 2 or 3. After that, we investigate more closely schemes of order 6 which possess non-normal closed subsets and non-commutative schemes of order 6 which possess a symmetric normal closed subset of order 3. (Non-commutative schemes of order 6 which possess a non-symmetric normal closed subset of order 3 or a normal closed subset of order 2 will be investigated in a forthcoming article.) In both cases, we explicitly give all irreducible (complex) representations of such schemes. Among other structural consequences we obtain that association schemes of order 6 are Coxeter schemes if they have two di erent non-normal closed subsets and that they are semidirect products if they possess a normal and a non-normal closed subset or a thin non-normal closed subset.