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 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.
