Projective modular representation theory of finite groups.

摘要

In this thesis, the author introduces the fundamental concept of projective Brauer characters for the first time and establishes projective Brauer theory or projective modular representation theory as a natural generalization for Brauer theory or modular representation theory of finite groups.;In Chapter 1, the author summarizes some basic results on projective representation theory of finite groups, with the aim of applying them in our new theory. To facilitate the understanding of the reader, the author also gives the necessary explanations. Except for three observations, all results in the first chapter are directly selected from the major reference [K1].;In Chapter 2, the author defines and studies projective Brauer characters as the starting point of our new theory. The crucial point is to use a pair of standard cocycles of p'-th roots of unity, in the respective base fields of the traditional p-modular system. They result in two sets of irreducible alpha-characters: alpha- Irr(G) and alpha-IBr( G). The author proves that irreducible Brauer alpha-characters are also C -linear independent on their domain G♭ , the alpha-regular part of p' -elements of G. The author also defines and studies alpha-decomposition matrix and Cartan alpha-matrix associated to irreducible Frobenius and Brauer alpha-characters.;In Chapter 3, the author defines and studies alpha-blocks of the above defined irreducible alpha-characters via central characters induced from projective irreducible characters. Then author derives block idempotents from primitive idempotents associated to irreducible Frobenius alpha-characters. Finally the author studies the properties of the associated block ideals.;In Chapter 4, the author first studies defect groups of alpha-regular classes and defect groups of alpha-blocks, then establishes the min-max theorem as a basic tool in this approach. Next the author defines Brauer homomorphisms between certain twisted group algebras. After a series of technical lemmas, the author proves the first main theorem as a major result of the thesis.