Block fusion systems and the center of the group ring

PhD Defense:

Ved Martin Wedel Jacobsen

Abstract:

This thesis develops some aspects of the theory of block fusion systems. Chapter 1 contains a brief introduction to the group algebra and some simple results about algebras over a field of positive characteristic. In chapter 2 we define the concept of a fusion system and the fundamental property of saturation. We also define block fusion systems and prove that they are saturated. Chapter 3 develops some tools for relating block fusion systems to the structure of the center of the group algebra. In particular, it is proven that a block has trivial defect group if and only if the center of the block algebra is one-dimensional. Chapter 4 consists of a proof that block fusion systems of symmetric groups are always group fusion systems of symmetric groups, and an analogous result holds for the alternating groups.

Resume:

Denne afhandling udvilker nogle aspekter af teorien for blokfusionssystemer. Kapitel 1 indeholder en kort introduktion til gruppealgebraen og nogle simple resultater om algebraer over et legeme med positiv karakteristik.

I kapitel 2 definerer vi konceptet et fusionssystem og den grundlæggendeegenskab mættethed. Vi definerer ogs°a blokfusionssystemer og beviser at deer mættede. Kapitel 3 udvikler nogle værktøjer til at relatere blokfusionssystemer til strukturen af gruppealgebraens center. Specielt bevises det at en blok har triviel defektgruppe hvis og kun hvis blokalgebraens center er en-dimensionalt. Kapitel 4 best°ar af et bevis for at blokfusionssystemer for symmetriske grupper altid er gruppefusionssystemer for symmetriske grupper, og et analogt resultat gælder for de alternerende grupper.

Principal supervisor: Prof. Jesper Michael Møller, University of Copenhagen

Assessement committee:

Prof. Jørn Børling Olsson, (Chairman), University of Copenhagen

Prof. Radha Kessar,  City University London

Ass. Prof. Radu Stancu, Universite de Picardie