Representation Theory

1. Front Matter

   Pages i-xv

   PDF

2. Finite Groups

   1. Front Matter

      Pages 1-2

      PDF

   2. Representations of Finite Groups

      • William Fulton, Joe Harris

      Pages 3-11

   3. Characters

      • William Fulton, Joe Harris

      Pages 12-25

   4. Examples; Induced Representations; Group Algebras; Real Representations

      • William Fulton, Joe Harris

      Pages 26-43

   5. Representations of\({\mathfrak{S}_{_d}}\): Young Diagrams and Frobenius’s Character Formula

      • William Fulton, Joe Harris

      Pages 44-62

   6. Representations of\({\mathfrak{U}_d}\)and\(G{L_2}\left( {{\mathbb{F}_q}} \right)\)

      • William Fulton, Joe Harris

      Pages 63-74

   7. Weyl’s Construction

      • William Fulton, Joe Harris

      Pages 75-88

3. Lie Groups and Lie Algebras

   1. Front Matter

      Pages 89-91

      PDF

   2. Lie Groups

      • William Fulton, Joe Harris

      Pages 93-103

   3. Lie Algebras and Lie Groups

      • William Fulton, Joe Harris

      Pages 104-120

   4. Initial Classification of Lie Algebras

      • William Fulton, Joe Harris

      Pages 121-132

   5. Lie Algebras in Dimensions One, Two, and Three

      • William Fulton, Joe Harris

      Pages 133-145

   6. Representations of sl2ℂ

      • William Fulton, Joe Harris

      Pages 146-160

   7. Representations of sl3ℂ, Part I

      • William Fulton, Joe Harris

      Pages 161-174

   8. Representations ofsl3ℂ, Part II: Mainly Lots of Examples

      • William Fulton, Joe Harris

      Pages 175-193

4. The Classical Lie Algebras and Their Representations

   1. Front Matter

      Pages 195-195

      PDF

   2. The General Setup: Analyzing the Structure and Representations of an Arbitrary Semisimple Lie Algebra

      • William Fulton, Joe Harris

      Pages 197-210

   3. sl4ℂ and slnℂ

      • William Fulton, Joe Harris

      Pages 211-237

   4. Symplectic Lie Algebras

      • William Fulton, Joe Harris

      Pages 238-252

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.

• Abelian group
• algebra
• cohomology
• cohomology group
• finite group
• group action
• homology
• Lie algebra
• lie group
• representation theory
• Vector space

• Department of Mathematics, University of Michigan, Ann Arbor, USA

  William Fulton

• Department of Mathematics, Harvard University, Cambridge, USA

  Joe Harris

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