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Table of Contents
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 Setup: 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.