Browsing by Subject "Burnside rings."
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Item Unknown Fusion systems in group representation theory(2013) Tuvay, İpekResults 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.Item Unknown The monomial Burnside functor(2009) Okay, CihanGiven 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}.Item Unknown On monomial Burnside rings(2003) Yaraneri, ErgünThis thesis is concerned with some different aspects of the monomial Burnside rings, including an extensive, self contained introduction of the A−fibred G−sets, and the monomial Burnside rings. However, this work has two main subjects that are studied in chapters 6 and 7. There are certain important maps studied by Yoshida in [16] which are very helpful in understanding the structure of the Burnside rings and their unit groups. In chapter 6, we extend these maps to the monomial Burnside rings and find the images of the primitive idempotents of the monomial Burnside C−algebras. For two of these maps, the images of the primitive idempotents appear for the first time in this work. In chapter 7, developing a line of research persued by Dress [9], Boltje [6], Barker [1], we study the prime ideals of monomial Burnside rings, and the primitive idempotents of monomial Burnside algebras. The new results include; (a): If A is a π−group, then the primitive idempotents of Z(π)B(A, G) and Z(π)B(G) are the same (b): If G is a π 0−group, then the primitive idempotents of Z(π)B(A, G) and QB(A, G) are the same (c): If G is a nilpotent group, then there is a bijection between the primitive idempotents of Z(π)B(A, G) and the primitive idempotents of QB(A, K) where K is the unique Hall π 0−subgroup of G. (Z(π) = {a/b ∈ Q : b /∈ ∪p∈πpZ}, π =a set of prime numbers).Item Unknown Real monomial Burnside rings and a decomposition of the the tom Dieck map(2009) Tuvay, İpekThis thesis is mainly concerned with a decomposition of the reduced tom Dieck map die : f A(RG) → B(G) × into two maps die+ and die− of the real monomial Burnside ring. The key idea is to introduce a real Lefschetz invariant as an element of the real monomial Burnside ring and to generalize the assertion that the image of an RG-module under the tom Dieck map coincides with the Lefschetz invariant of the sphere of the same module.