Clifford Theory for Group Representations

Abstract

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.