Clifford Theory for Group RepresentationsBy
- G. Karpilovsky
Let N be a normal subgroup of a finite group G and let F be a field. An important method for constructing irreducible FG-modules consists of the application (perhaps repeated) of three basic operations: (i) restriction to FN. (ii) extension from FN. (iii) induction from FN. This is the `Clifford Theory' developed by Clifford in 1937. In the past twenty years, the theory has enjoyed a period of vigorous development. The foundations have been strengthened and reorganized from new points of view, especially from the viewpoint of graded rings and crossed products.The purpose of this monograph is to tie together various threads of the development in order to give a comprehensive picture of the current state of the subject. It is assumed that the reader has had the equivalent of a standard first-year graduate algebra course, i.e. familiarity with basic ring-theoretic, number-theoretic and group-theoretic concepts, and an understanding of elementary properties of modules, tensor products and fields.
North-Holland Mathematics Studies
Published: May 1989
- Preliminaries. Matrix Rings. Artinian, Noetherian and Completely Reducible Modules. The Radical of Modules and Rings. Unique Decompositions. Group Algebras. Cohomology Groups. Restriction to Normal Subgroups. Induced and Relatively Projective Modules. Restriction of Irreducible Modules to Normal Subgroups. Lifting Idempotents. Restriction of Indecomposable Modules to Normal Subgroups. Similarity with Ground Field Extensions: Preliminary Results, Indecomposable Modules, Irreducible Modules. Restriction of Projective Covers. Induction and Extension from Normal Subgroups. Group-Graded Algebras and Crossed Products. Graded Ideals. The Endomorphism Ring of Induced Modules. Inducing G-Invariant Modules. Indecomposability of Induced Modules. Homogeneity of Induced Modules. Homogeneity of Induced Modules: An Alternative Approach. The Loewy Length of Induced Modules. Extension from Normal Subgroups: Basic Constructions. Counting Nonisomorphic Extensions. Projective Representations and Inflated Modules. Restriction and Induction of Absolutely Irreducible Modules. Applications: Dimensions of Irreducible Modules and Their Projective Covers. Extensions of Modules over Arbitrary Fields: Projective Crossed Representations, Obstructions to Extensions, The Isaacs-Dade Theorem. Bibliography. Index.