Representation Theory: A First Course (Hardcover)

I: Finite Groups.- 1. Representations of Finite Groups.- 1.1: Definitions.- 1.2: Complete Reducibility; Schur's Lemma.- 1.3: Examples: Abelian Groups; $$ {\mathfrak{S}_3}$$.- 2. Characters.- 2.1: Characters.- 2.2: The First Projection Formula and Its Consequences.- 2.3: Examples: $$ {\mathfrak{S}_4}$$ and $$ {\mathfrak{A}_4}$$.- 2.4: More Projection Formulas; More Consequences.- 3. Examples; Induced Representations; Group Algebras; Real Representations.- 3.1: Examples: $$ {\mathfrak{S}_5}$$ and $$ {\mathfrak{A}_5}$$.- 3.2: Exterior Powers of the Standard Representation of $$ {\mathfrak{S}_d}$$.- 3.3: Induced Representations.- 3.4: The Group Algebra.- 3.5: Real Representations and Representations over Subfields of $$ \mathbb{C}$$.- 4. Representations of: $$ {\mathfrak{S}_d}$$ Young Diagrams and Frobenius's Character Formula.- 4.1: Statements of the Results.- 4.2: Irreducible Representations of $$ {\mathfrak{S}_d}$$.- 4.3: Proof of Frobenius's Formula.- 5. Representations of $$ {\mathfrak{A}_d}$$ and $$ G{L_2}\left( {{\mathbb{F}_q}} ight)$$.- 5.1: Representations of $$ {\mathfrak{A}_d}$$.- 5.2: Representations of $$ G{L_2}\left( {{\mathbb{F}_q}} ight)$$ and $$ S{L_2}\left( {{\mathbb{F}_q}} ight)$$.- 6. Weyl's Construction.- 6.1: Schur Functors and Their Characters.- 6.2: The Proofs.- II: Lie Groups and Lie Algebras.- 7. Lie Groups.- 7.1: Lie Groups: Definitions.- 7.2: Examples of Lie Groups.- 7.3: Two Constructions.- 8. Lie Algebras and Lie Groups.- 8.1: Lie Algebras: Motivation and Definition.- 8.2: Examples of Lie Algebras.- 8.3: The Exponential Map.- 9. Initial Classification of Lie Algebras.- 9.1: Rough Classification of Lie Algebras.- 9.2: Engel's Theorem and Lie's Theorem.- 9.3: Semisimple Lie Algebras.- 9.4: Simple Lie Algebras.- 10. Lie Algebras in Dimensions One, Two, and Three.- 10.1: Dimensions One and Two.- 10.2: Dimension Three, Rank 1.- 10.3: Dimension Three, Rank 2.- 10.4: Dimension Three, Rank 3.- 11. Representations of $$ \mathfrak{s}{\mathfrak{l}_2}\mathbb{C}$$.- 11.1: The Irreducible Representations.- 11.2: A Little Plethysm.- 11.3: A Little Geometric Plethysm.- 12. Representations of $$ \mathfrak{s}{\mathfrak{l}_3}\mathbb{C},$$ Part I.- 13. Representations of $$ \mathfrak{s}{\mathfrak{l}_3}\mathbb{C},$$ Part II: Mainly Lots of Examples.- 13.1: Examples.- 13.2: Description of the Irreducible Representations.- 13.3: A Little More Plethysm.- 13.4: A Little More Geometric Plethysm.- III: The Classical Lie Algebras and Their Representations.- 14. The General Set-up: Analyzing the Structure and Representations of an Arbitrary Semisimple Lie Algebra.- 14.1: Analyzing Simple Lie Algebras in General.- 14.2: About the Killing Form.- 15. $$ \mathfrak{s}{\mathfrak{l}_4}\mathbb{C}$$ and $$ \mathfrak{s}{\mathfrak{l}_n}\mathbb{C}$$.- 15.1: Analyzing $$ \mathfrak{s}{\mathfrak{l}_n}\mathbb{C}$$.- 15.2: Representations of $$ \mathfrak{s}{\mathfrak{l}_4}\mathbb{C}$$ and $$ \mathfrak{s}{\mathfrak{l}_n}\mathbb{C}$$.- 15.3: Weyl's Construction and Tensor Products.- 15.4: Some More Geometry.- 15.5: Representations of $$ G{L_n}\mathbb{C}$$.- 16. Symplectic Lie Algebras.- 16.1: The Structure of $$S{p_{2n}}\mathbb{C}$$ and $$ \mathfrak{s}{\mathfrak{p}_2n}\mathbb{C}$$.- 16.2: Representations of $$ \mathfrak{s}{\mathfrak{p}_4}\mathbb{C}$$.- 17. $$ \mathfrak{s}{\mathfrak{p}_6}\mathbb{C}$$ and $$ \mathfrak{s}{\mathfrak{p}_2n}\mathbb{C}$$.- 17.1: Representations of $$ \mathfrak{s}{\mathfrak{p}_6}\mathbb{C}$$.- 17.2: Representations of $$ \mathfrak{s}{\mathfrak{p}_2n}\mathbb{C}$$ in General.- 17.3: Weyl's Construction for Symplectic Groups.- 18. Orthogonal Lie Algebras.- 18.1: $$ S{O_m}\mathbb{C}$$ and $$ \mathfrak{s}{\mathfrak{o}_m}\mathbb{C}$$.- 18.2: Representations of $$ \mathfrak{s}{\mathfrak{o}_3}\mathbb{C},$$$$ \mathfrak{s}{\mathfrak{o}_4}\mathbb{C},$$ and $$ \mathfrak{s}{\mathfrak{o}_5}\mathbb