17 research outputs found

    Fusion systems on bicyclic 2-groups

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    We classify all (saturated) fusion systems on bicyclic 2-groups. Here, a bicyclic group is a product of two cyclic subgroups. This extends previous work on fusion systems on metacyclic 2-groups (see [Craven-Glesser, 2012] and [Sambale, 2012]). As an application we prove Olsson's Conjecture for all blocks with bicyclic defect groups.Comment: 22 pages, shorted and some arguments replace

    2-Groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs

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    Buildings, Group Homology and Lattices

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    This is the author's PhD thesis, published at the Universit\"at M\"unster, Germany in 2010. It contains a detailed description of the results of arXiv:0903.1989, arXiv:0905.0071 and arXiv:0908.2713.Comment: 171 pages, PhD thesis, Universit\"at M\"unster, see http://nbn-resolving.de/urn:nbn:de:hbz:6-1748954938

    Categorical post-quantum theories

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    In this thesis of two Parts, we investigate the application of categorical methods to modelling post-quantum theories. In Part I we study hyper-decoherence between quantum-like theories. Chapter 1 serves as an introduction to Categorical Probabilistic Theories which combine elements of Categorical Quantum Mechanics and Operational Probabilistic Theories, and to CPM categories which generalise the CPM construction of Selinger to allow for richer group symmetries. In Chapter 2 we study the theory of density hypercubes which exhibits a hyper-decoherence mechanism witnessing quantum theory as an effectful subtheory. We show that this hyper-decoherence process is probabilistic within the theory of density hypercubes and discuss some plausible operational interpretations of this. As a result, we side-step a no-go result regarding the existence of deterministic hyper-decoherence maps, showing that it is nevertheless possible for a post-quantum theory to possess probabilistic hyper-decoherence maps. In Chapter 3 we focus on a particular case of the CPM construction, where the symmetries are generated by the Galois group of a finite field extension. We discuss how to construct probabilistic theories which form towers of decoherence in bijection with the subfields of a Galois extension. These towers generalise the decoherence process of standard quantum theory. In Part II we study profunctorial methods and their application to spacetime and quantum supermaps. Chapter 4 serves as an introduction to profunctors, promonoidal categories and premonoidal categories, including the enriched version of the latter. Chapter 5 introduces some toy categories of causal curves in spacetime and discusses how we might upgrade the partial monoidal structure of such categories to a total tensor using both pre- and promonoidal categories. Chapter 6 makes this combination of pre- and promonoidal categories more formal, introducing the notion of a pro-effectful category. In the final Chapter 7 we describe how we can use the category of coend optics as a model of quantum combs. We describe the promonoidal structures on this category and their interpretation as horizontal and vertical composition of holes in monoidal categories. We also generalise coend optics to allow for a premonoidal base category, and point towards how the methods of this Chapter might be extended to include arbitrary quantum supermaps

    36th International Symposium on Theoretical Aspects of Computer Science: STACS 2019, March 13-16, 2019, Berlin, Germany

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