The monomial Burnside functor

Ankara : The Department of Mathematics and the Institute of Engineering and Science of Bilkent University, 2009.Thesis (Master's) -- Bilkent University, 2009.Includes bibliographical references leaves 29.Given a finite group G, we can realize the permutation modules by the linearization map defined from the Burnside ring B(G) to the character ring of G, denoted AK(G). But not all KG-modules are permutation modules. To realize all the KGmodules we need to replace B(G) by the monomial Burnside ring BC(G). We can get information about monomial Burnside ring of G by considering subgroups or quotient groups of G. For this the setting of biset functors is suitable. We can consider the monomial Burnside ring as a biset functor and study the elemental maps: transfer, retriction, inflation, deflation and isogation. Among these maps, deflation is somewhat difficult and requires more consideration. In particular, we examine deflation for p-groups and study the simple composition factors of the monomial Burnside functor for 2-groups with the fibre group {±1}.Okay, CihanM.S

Simplicial distributions, convex categories and contextuality

The data of a physical experiment can be represented as a presheaf of probability distributions. A striking feature of quantum theory is that those probability distributions obtained in quantum mechanical experiments do not always admit a joint probability distribution, a celebrated observation due to Bell. Such distributions are called contextual. Simplicial distributions are combinatorial models that extend presheaves of probability distributions by elevating sets of measurements and outcomes to spaces. Contextuality can be defined in this generalized setting. This paper introduces the notion of convex categories to study simplicial distributions from a categorical perspective. Simplicial distributions can be given the structure of a convex monoid, a convex category with a single object, when the outcome space has the structure of a group. We describe contextuality as a monoid-theoretic notion by introducing a weak version of invertibility for monoids. Our main result is that a simplicial distribution is noncontextual if and only if it is weakly invertible. Similarly, strong contextuality and contextual fraction can be characterized in terms of invertibility in monoids. Finally, we show that simplicial homotopy can be used to detect extremal simplicial distributions refining the earlier methods based on Cech cohomology and the cohomology of groups.Comment: 42 pages, 4 figure

Equivariant simplicial distributions and quantum contextuality

We introduce an equivariant version of contextuality with respect to a symmetry group, which comes with natural applications to quantum theory. In the equivariant setting, we construct cohomology classes that can detect contextuality. This framework is motivated by the earlier topological approach to contextuality producing cohomology classes that serve as computational primitives in measurement-based quantum computing.Comment: 36 pages, 4 figure