Representation Theory by Harris, Joe -- Read -- Imperial Library of Trantor

Index

Cover Title Page Copyright Page Preface Using This Book Table of Contents Part I Finite Groups

Lecture 1 Representations of Finite Groups

§1.1. Definitions §1.2. Complete Reducibility; Schur’s Lemma §1.3. Examples: Abelian Groups;

Lecture 2 Characters

§2.1. Characters §2.2. The First Projection Formula and Its Consequences §2.3. Examples: and §2.4. More Projection Formulas; More Consequences

Lecture 3 Examples; Induced Representations; Group Algebras; Real Representations

§3.1. Examples: and §3.2. Exterior Powers of the Standard Representation of §3.3. Induced Representations §3.4. The Group Algebra §3.5. Real Representations and Representations over Subfields of ℂ

Lecture 4 Representations of: Young Diagrams and Frobenius’s Character Formula

§4.1. Statements of the Results §4.2. Irreducible Representations of §4.3. Proof of Frobenius’s Formula

Lecture 5 Representations of and

§5.1. Representations of §5.2. Representations of and

Lecture 6 Weyl’s Construction

§6.1. Schur Functors and Their Characters §6.2. The Proofs

Part II Lie Groups and Lie Algebras

Lecture 7 Lie Groups

§7.1. Lie Groups: Definitions §7.2. Examples of Lie Groups §7.3. Two Constructions

Lecture 8 Lie Algebras and Lie Groups

§8.1. Lie Algebras: Motivation and Definition §8.2. Examples of Lie Algebras §8.3. The Exponential Map

Lecture 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

Lecture 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

Lecture 11 Representations of

§11.1. The Irreducible Representations §11.2. A Little Plethysm §11.3. A Little Geometric Plethysm

Lecture 12 Representations of , Part I Lecture 13 Representations of , 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

Part III The Classical Lie Algebras and Their Representations

Lecture 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

Lecture 15 and

§15.1. Analyzing §15.2. Representations of and §15.3. Weyl’s Construction and Tensor Products §15.4. Some More Geometry §15.5. Representations of GLnℂ

Lecture 16 Symplectic Lie Algebras

§16.1. The Structure of Sp2nℂ and §16.2. Representations of

Lecture 17 and

§17.1. Representations of §17.2. Representations of in General §17.3. Weyl’s Construction for Symplectic Groups

Lecture 18 Orthogonal Lie Algebras

§18.1. SOmℂ and §18.2. Representations of , , and

Lecture 19 and

§19.1. Representations of §19.2. Representations of the Even Orthogonal Algebras §19.3. Representations of §19.4. Representations of the Odd Orthogonal Algebras §19.5. Weyl’s Construction for Orthogonal Groups

Lecture 20 Spin Representations of

§20.1. Clifford Algebras and Spin Representations of §20.2. The Spin Groups Spinmℂ and Spinmℝ §20.3. Spin8ℂ and Triality

Part IV Lie Theory

Lecture 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

Lecture 22 and Other Exceptional Lie Algebras

§22.1. Construction of from Its Dynkin Diagram §22.2. Verifying That Is a Lie Algebra §22.3. Representations of §22.4. Algebraic Constructions of the Exceptional Lie Algebras

Lecture 23 Complex Lie Groups; Characters

§23.1. Representations of Complex Simple Lie Groups §23.2. Representation Rings and Characters §23.3. Homogeneous Spaces §23.4. Bruhat Decompositions

Lecture 24 Weyl Character Formula

§24.1. The Weyl Character Formula §24.2. Applications to Classical Lie Algebras and Groups

Lecture 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

Lecture 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 APPENDIX A On Symmetric Functions APPENDIX B On Multilinear Algebra APPENDIX C On Semisimplicity APPENDIX D Cartan Subalgebras APPENDIX E Ado’s and Levi’s Theorems APPENDIX F Invariant Theory for the Classical Groups Bibliography

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