Fusion systems in group representation theory

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.