Representation theory : a first course

William Fulton, Joe Harris

Introducing finite-dimensional representations of Lie groups and Lie algebras, this example-oriented book works from representation theory of finite groups, through Lie groups and Lie algrbras to the finite dimensional representations of the classical groups.

「Nielsen BookData」より

[目次]

• I: Finite Groups.- 1. Representations of Finite Groups.- x1.1: Definitions.- x1.2: Complete Reducibility
• Schur's Lemma.- x1.3: Examples: Abelian Groups
• $${\mathfrak{S}_3}$$.- 2. Characters.- x2.1: Characters.- x2.2: The First Projection Formula and Its Consequences.- x2.3: Examples: $${\mathfrak{S}_4}$$ and $${\mathfrak{A}_4}$$.- x2.4: More Projection Formulas
• More Consequences.- 3. Examples
• Induced Representations
• Group Algebras
• Real Representations.- x3.1: Examples: $${\mathfrak{S}_5}$$ and $${\mathfrak{A}_5}$$.- x3.2: Exterior Powers of the Standard Representation of $${\mathfrak{S}_d}$$.- x3.3: Induced Representations.- x3.4: The Group Algebra.- x3.5: Real Representations and Representations over Subfields of $$\mathbb{C}$$.- 4. Representations of: $${\mathfrak{S}_d}$$ Young Diagrams and Frobenius's Character Formula.- x4.1: Statements of the Results.- x4.2: Irreducible Representations of $${\mathfrak{S}_d}$$.- x4.3: Proof of Frobenius's Formula.- 5. Representations of $${\mathfrak{A}_d}$$ and $$G{L_2}\left( {{\mathbb{F}_q}} \right)$$.- x5.1: Representations of $${\mathfrak{A}_d}$$.- x5.2: Representations of $$G{L_2}\left( {{\mathbb{F}_q}} \right)$$ and $$S{L_2}\left( {{\mathbb{F}_q}} \right)$$.- 6. Weyl's Construction.- x6.1: Schur Functors and Their Characters.- x6.2: The Proofs.- II: Lie Groups and Lie Algebras.- 7. Lie Groups.- x7.1: Lie Groups: Definitions.- x7.2: Examples of Lie Groups.- x7.3: Two Constructions.- 8. Lie Algebras and Lie Groups.- x8.1: Lie Algebras: Motivation and Definition.- x8.2: Examples of Lie Algebras.- x8.3: The Exponential Map.- 9. Initial Classification of Lie Algebras.- x9.1: Rough Classification of Lie Algebras.- x9.2: Engel's Theorem and Lie's Theorem.- x9.3: Semisimple Lie Algebras.- x9.4: Simple Lie Algebras.- 10. Lie Algebras in Dimensions One, Two, and Three.- x10.1: Dimensions One and Two.- x10.2: Dimension Three, Rank 1.- x10.3: Dimension Three, Rank 2.- x10.4: Dimension Three, Rank 3.- 11. Representations of $$\mathfrak{s}{\mathfrak{l}_2}\mathbb{C}$$.- x11.1: The Irreducible Representations.- x11.2: A Little Plethysm.- x11.3: 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.- x13.1: Examples.- x13.2: Description of the Irreducible Representations.- x13.3: A Little More Plethysm.- x13.4: 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.- x14.1: Analyzing Simple Lie Algebras in General.- x14.2: About the Killing Form.- 15. $$\mathfrak{s}{\mathfrak{l}_4}\mathbb{C}$$ and $$\mathfrak{s}{\mathfrak{l}_n}\mathbb{C}$$.- x15.1: Analyzing $$\mathfrak{s}{\mathfrak{l}_n}\mathbb{C}$$.- x15.2: Representations of $$\mathfrak{s}{\mathfrak{l}_4}\mathbb{C}$$ and $$\mathfrak{s}{\mathfrak{l}_n}\mathbb{C}$$.- x15.3: Weyl's Construction and Tensor Products.- x15.4: Some More Geometry.- x15.5: Representations of $$G{L_n}\mathbb{C}$$.- 16. Symplectic Lie Algebras.- x16.1: The Structure of $$S{p_{2n}}\mathbb{C}$$ and $$\mathfrak{s}{\mathfrak{p}_2n}\mathbb{C}$$.- x16.2: Representations of $$\mathfrak{s}{\mathfrak{p}_4}\mathbb{C}$$.- 17. $$\mathfrak{s}{\mathfrak{p}_6}\mathbb{C}$$ and $$\mathfrak{s}{\mathfrak{p}_2n}\mathbb{C}$$.- x17.1: Representations of $$\mathfrak{s}{\mathfrak{p}_6}\mathbb{C}$$.- x17.2: Representations of $$\mathfrak{s}{\mathfrak{p}_2n}\mathbb{C}$$ in General.- x17.3: Weyl's Construction for Symplectic Groups.- 18. Orthogonal Lie Algebras.- x18.1: $$S{O_m}\mathbb{C}$$ and $$\mathfrak{s}{\mathfrak{o}_m}\mathbb{C}$$.- x18.2: 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 $$\mathfrak{s}{\mathfrak{o}_m}\mathbb{C}$$.- x19.1: Representations of $$\mathfrak{s}{\mathfrak{o}_6}\mathbb{C}$$.- x19.2: Representations of the Even Orthogonal Algebras.- x19.3: Representations of $$\mathfrak{s}{\mathfrak{o}_7}\mathbb{C}$$.- x19.4. Representations of the Odd Orthogonal Algebras.- x19.5: Weyl's Construction for Orthogonal Groups.- 20. Spin Representations of $$\mathfrak{s}{\mathfrak{o}_m}\mathbb{C}$$.- x20.1: Clifford Algebras and Spin Representations of $$\mathfrak{s}{\mathfrak{o}_m}\mathbb{C}$$.- x20.2: The Spin Groups $$Spi{n_m}\mathbb{C}$$ and $$Spi{n_m}\mathbb{R}$$.- x20.3: $$Spi{n_8}\mathbb{C}$$ and Triality.- IV: Lie Theory.- 21. The Classification of Complex Simple Lie Algebras.- x21.1: Dynkin Diagrams Associated to Semisimple Lie Algebras.- x21.2: Classifying Dynkin Diagrams.- x21.3: Recovering a Lie Algebra from Its Dynkin Diagram.- 22. $${g_2}$$and Other Exceptional Lie Algebras.- x22.1: Construction of $${g_2}$$ from Its Dynkin Diagram.- x22.2: Verifying That $${g_2}$$ is a Lie Algebra.- x22.3: Representations of $${{\mathfrak{g}}_{2}}$$.- x22.4: Algebraic Constructions of the Exceptional Lie Algebras.- 23. Complex Lie Groups
• Characters.- x23.1: Representations of Complex Simple Groups.- x23.2: Representation Rings and Characters.- x23.3: Homogeneous Spaces.- x23.4: Bruhat Decompositions.- 24. Weyl Character Formula.- x24.1: The Weyl Character Formula.- x24.2: Applications to Classical Lie Algebras and Groups.- 25. More Character Formulas.- x25.1: Freudenthal's Multiplicity Formula.- x25.2: Proof of (WCF)
• the Kostant Multiplicity Formula.- x25.3: Tensor Products and Restrictions to Subgroups.- 26. Real Lie Algebras and Lie Groups.- x26.1: Classification of Real Simple Lie Algebras and Groups.- x26.2: Second Proof of Weyl's Character Formula.- x26.3: Real, Complex, and Quaternionic Representations.- Appendices.- A. On Symmetric Functions.- xA.1: Basic Symmetric Polynomials and Relations among Them.- xA.2: Proofs of the Determinantal Identities.- xA.3: Other Determinantal Identities.- B. On Multilinear Algebra.- xB.1: Tensor Products.- xB.2: Exterior and Symmetric Powers.- xB.3: Duals and Contractions.- C. On Semisimplicity.- xC.1: The Killing Form and Caftan's Criterion.- xC.2: Complete Reducibility and the Jordan Decomposition.- xC.3: On Derivations.- D. Cartan Subalgebras.- xD.1: The Existence of Cartan Subalgebras.- xD.2: On the Structure of Semisimple Lie Algebras.- xD.3: The Conjugacy of Cartan Subalgebras.- xD.4: On the Weyl Group.- E. Ado's and Levi's Theorems.- xE.1: Levi's Theorem.- xE.2: Ado's Theorem.- F. Invariant Theory for the Classical Groups.- xF.1: The Polynomial Invariants.- xF.2: Applications to Symplectic and Orthogonal Groups.- xF.3: Proof of Capelli's Identity.- Hints, Answers, and References.- Index of Symbols.

「Nielsen BookData」より

書名 Representation theory : a first course Fulton, William Harris, Joe Graduate texts in mathematics Springer-Verlag 1999, c1991 5th corr. printing xv, 551 p. 24 cm 0387974954 3540974954 3540975276 0387975276 BA49505082 ※クリックでCiNii Booksを表示 英語 アメリカ合衆国
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