Browsing by Subject "Induction (Mathematics)"

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    Canonical induction for trivial source rings
    (2013) Büyükçolak, Yasemin
    We discuss the canonical induction formula for some special Mackey functors by following the construction of Boltje. These functors are the ordinary and modular character rings and the trivial source rings. Making use of a natural correspondence between the Mackey algebra and the finite algebra spanned by the three kinds of basic bisets, namely the conjugation, restriction and induction, we investigate the canonical induction formula in terms of the theory of bisets. We focus on the trivial source rings and the canonical induction formula for them. The main aim is to get an explicit formula for the canonical induction of regular bimodules in the trivial source. This gives a first step towards for the canonical induction of blocks.
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    A correspondence of simple alcahestic group functors
    (2008) Coşkun, Olcay
    Representation theory of finite groups associates two classical constructions to a group G, namely the representation ring of G and the Burnside ring of G. These rings share a special structure that comes from three classical maps, namely restriction, conjugation, and transfer maps. These are not the only objects having this structure and the theory of Mackey functors, introduced by Green, unifies the treatment of such objects. The above constructions share a further structure that comes from two other maps, the inflation map and the deflation map. Unified treatment of the objects having this further structure was introduced by Bouc [4]. These objects are called biset functors. Between Mackey functors and biset functors there lies more natural constructions, for example the functor of group (co)homology. In order to handle these intermediate structures, Bouc introduced another concept, now known as globallydefined Mackey functors, a name given by Webb. In this thesis, we unify the above theories by considering the algebra whose module category is equivalent to the category of biset functors and by introducing alcahestic group functors. Our main results classify and describe simple alcahestic group functors and give a criterion of semisimplicity for the categories of these functors.
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    Mackey group categories and their simple functors
    (2012) Yaylıoğlu, Volkan Dağhan
    Constructing the Mackey group category M using axioms which are reminiscent of fusion systems, the simple RM-functors (the simple functors from the R-linear extension of M to R-modules, where R is a commutative ring) can be classified via pairs consisting of the objects of the Mackey group category (which are finite groups) and simple modules of specific group algebras. The key ingredient to this classification is a bijection between some RM-functors (not necessarily simple) and some morphisms of EndRM(G). It is also possible to define the Mackey group category by using Brauer pairs, or even pointed groups as objects so that this classification will still be valid.