Browsing by Subject "semisimplicity"

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Open Access
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.
Open Access
Fusion systems in group representation theory
(2013) Tuvay, İpek
Results on the Mackey category MF corresponding to a fusion system F and fusion systems defined on p-permutation algebras are our main concern. In the first part, we give a new proof of semisimplicity of MF over C by using a different method than the method used by Boltje and Danz. Following their work in [8], we construct the ghost algebra corresponding to the quiver algebra of MF which is isomorphic to the quiver algebra. We then find a formula for the centrally primitive mutually orthogonal idempotents of this ghost algebra. Then we use this formula to give an alternative proof of semisimplicity of the quiver algebra of MF over the complex numbers. In the second part, we focus on finding classes of p-permutation algebras which give rise to a saturated fusion system which has been studied by Kessar-KunugiMutsihashi in [16]. By specializing to a particular p-permutation algebra and using a result of [16], the question is reduced to finding Brauer indecomposable p-permutation modules. We show for some particular cases of fusion systems we have Brauer indecomposability. In the last part, we study real representations using the real monomial Burnside ring. We deduce a relation on the dimensions of the subgroup-fixed subspaces of a real representation.