Representation Theory A First Course

The primary goal of these lectures is to introduce a beginner to the finite­ dimensional representations of Lie groups and Lie algebras. Since this goal is shared by quite a few other books, we should explain in this Preface how our approach differs, although the potential reader can probably see this better by a quick browse through the book. Representation theory is simple to define: it is the study of the ways in which a given group may act on vector spaces. It is almost certainly unique, however, among such clearly delineated subjects, in the breadth of its interest to mathematicians. This is not surprising: group actions are ubiquitous in 20th century mathematics, and where the object on which a group acts is not a vector space, we have learned to replace it by one that is {e. g. , a cohomology group, tangent space, etc. }. As a consequence, many mathematicians other than specialists in the field {or even those who think they might want to be} come in contact with the subject in various ways. It is for such people that this text is designed. To put it another way, we intend this as a book for beginners to learn from and not as a reference. This idea essentially determines the choice of material covered here. As simple as is the definition of representation theory given above, it fragments considerably when we try to get more specific.

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.