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Tensor Products, Trivial Source Modules and Related Algebras
Christopher C. Gill Lincoln College University of Oxford
A thesis submitted for the degree of Doctor of Philosophy Michaelmas Term 2010
Abstract
In this thesis we study trivial source modules for group algebras. In particular, much of the thesis is devoted to the study of Young modules for symmetric groups. In chapter 1 we give background material that will be used in the thesis. Chapter 2 is devoted to studying the Young ring. We describe a complete set of primitive idempotents, prove some reduction formulas for the p-Kostka numbers, and give a method for calculating tensor products of Young modules in some cases. In the third chapter we classify the indecomposable Young permutation modules. In the case p = 2, we study the endomorphism algebra of the direct sum of all indecomposable Young permutation modules, and show that this behaves well under certain known embeddings of the Schur algebras. In chapter 4 we study tensor products of trivial source modules, and in particular we study the vertices of direct summands of such tensor products. We apply these results to Young modules and relating the general results to Young vertices. We then apply the results to Young modules with the same vertex, proving combinatorial results and reduction formulas for multiplicities of direct summands of such tensor products. The fifth chapter studies the periodic Young modules. We describe such Young modules combinatorially and determine their distribution into blocks. We determine the period of all periodic Young modules in any characteristic. In chapter 6 we study the Scott algebra for a finite group G. This is the endomorphism algebra of a direct sum of Scott modules for G. We determine the Cartan matrix in some cases, and describe some properties of the quiver of a Scott algebra for Hamiltonian p-groups.
Acknowledgements
I would like to thank my supervisor Dr Karin Erdmann for suggesting interesting problems for me, for her many useful suggestions, her patience, and also her careful reading of my work. I gratefully acknowledge the financial support of the Engineering and Physical Sciences Research Council. I would also like to thank all the people who shared an office with me at various times through the course of my studies: Dusko Bogdanic, Sarah Scherotzke, Armin Shalile, David Craven, Aram Mikaelian, and Sigurdur Hannesson. They have provided me with many useful and interesting discussions, and made my time in the Mathematical Institute enjoyable. Last, but certainly not least I thank my family for their help and support during the course of my education.
Contents
1 Introduction 1.1
1
Background material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1.1
Background on representations of finite groups . . . . . . . . . . . .
3
1.1.1.1
Tensor products . . . . . . . . . . . . . . . . . . . . . . . .
3
1.1.1.2
Induction and restriction . . . . . . . . . . . . . . . . . . .
4
1.1.1.3
Relative projectivity, vertices, sources and the Green correspondence
. . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.1.1.4
Projective covers, and minimal projective resolutions . . .
9
1.1.1.5
Introduction to permutation modules and their summands
10
1.1.1.6
Ordinary characters . . . . . . . . . . . . . . . . . . . . . .
12
1.1.1.7
The Brauer morphism and the Brou´e correspondence . . .
15
1.1.1.8
Scott modules . . . . . . . . . . . . . . . . . . . . . . . . .
17
Background on representations of symmetric groups . . . . . . . . .
19
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