## Fusion systems in group representation theory

##### Author

Tuvay, İpek

##### Advisor

Barker, Laurence J.

##### Date

2013##### Publisher

Bilkent University

##### Language

English

##### Type

Thesis##### Item Usage Stats

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Show full item record##### Abstract

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.

##### Keywords

fusion systemMackey category

semisimplicity

p-permutation algebra

Brauer indeomposability

monomial Burnside ring