Representation Theory: A First Course

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$$G{L_2}\left( {{\mathbb{F}_q}} \right)$$.- 5.1: Representations of$${\mathfrak{A}_d}$$.- 5.2: Representations of$$G{L_2}\left( {{\mathbb{F}_q}} \right)$$and$$S{L_2}\left( {{\mathbb{F}_q}} \right)$$.- 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$$\mathfrak{s}{\mathfrak{o}_5}\mathbb{C}$$.- 19.$$\mathfrak{s}{\mathfrak{o}_6}\mathbb{C},$$$$\mathfrak{s}{\mathfrak{o}_7}\mathbb{C},$$and$$\mathfrak{s}{\mathfrak{o}_m}\mathbb{C}$$.- 19.1: Representations of$$\mathfrak{s}{\mathfrak{o}_6}\mathbb{C}$$.- 19.2: Representations of the Even Orthogonal Algebras.- 19.3: Representations of$$\mathfrak{s}{\mathfrak{o}_7}\mathbb{C}$$.- 19.4. Representations of the Odd Orthogonal Algebras.- 19.5: Weyl's Construction for Orthogonal Groups.- 20. Spin Representations of$$\mathfrak{s}{\mathfrak{o}_m}\mathbb{C}$$.- 20.1: Clifford Algebras and Spin Representations of $$\mathfrak{s}{\mathfrak{o}_m}\mathbb{C}$$.- 20.2: The Spin Groups$$Spi{n_m}\mathbb{C}$$and$$Spi{n_m}\mathbb{R}$$.- 20.3:$$Spi{n_8}\mathbb{C}$$and Triality.- IV: Lie Theory.- 21. The Classification of Complex Simple Lie Algebras.- 21.1: Dynkin Diagrams Associated to Semisimple Lie Algebras.- 21.2: Classifying Dynkin Diagrams.- 21.3: Recovering a Lie Algebra from Its Dynkin Diagram.- 22. $${g_2}$$and Other Exceptional Lie Algebras.- 22.1: Construction of$${g_2}$$from Its Dynkin Diagram.- 22.2: Verifying That$${g_2}$$is a Lie Algebra.- 22.3: Representations of$${{\mathfrak{g}}_{2}}$$.- 22.4: Algebraic Constructions of the Exceptional Lie Algebras.- 23. Complex Lie Groups; Characters.- 23.1: Representations of Complex Simple Groups.- 23.2: Representation Rings and Characters.- 23.3: Homogeneous Spaces.- 23.4: Bruhat Decompositions.- 24. Weyl Character Formula.- 24.1: The Weyl Character Formula.- 24.2: Applications to Classical Lie Algebras and Groups.- 25. More Character Formulas.- 25.1: Freudenthal's Multiplicity Formula.- 25.2: Proof of (WCF); the Kostant Multiplicity Formula.- 25.3: Tensor Products and Restrictions to Subgroups.- 26. Real Lie Algebras and Lie Groups.- 26.1: Classification of Real Simple Lie Algebras and Groups.- 26.2: Second Proof of Weyl's Character Formula.- 26.3: Real, Complex, and Quaternionic Representations.- Appendices.- A. On Symmetric Functions.- A.1: Basic Symmetric Polynomials and Relations among Them.- A.2: Proofs of the Determinantal Identities.- A.3: Other Determinantal Identities.- B. On Multilinear Algebra.- B.1: Tensor Products.- B.2: Exterior and Symmetric Powers.- B.3: Duals and Contractions.- C. On Semisimplicity.- C.1: The Killing Form and Caftan's Criterion.- C.2: Complete Reducibility and the Jordan Decomposition.- C.3: On Derivations.- D. Cartan Subalgebras.- D.1: The Existence of Cartan Subalgebras.- D.2: On the Structure of Semisimple Lie Algebras.- D.3: The Conjugacy of Cartan Subalgebras.- D.4: On the Weyl Group.- E. Ado's and Levi's Theorems.- E.1: Levi's Theorem.- E.2: Ado's Theorem.- F. Invariant Theory for the Classical Groups.- F.1: The Polynomial Invariants.- F.2: Applications to Symplectic and Orthogonal Groups.- F.3: Proof of Capelli's Identity.- Hints, Answers, and References.- Index of Symbols.