Titlebook: Representation Theory; A First Course William Fulton,Joe Harris Textbook 2004 Springer Science+Business Media New York 2004 Abelian group.a

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Representations of,: Young Diagrams and Frobenius’s Character Formulas, a construction of the representations (via Young symmetrizers) and a formula (Frobenius’ formula) for their characters. The proof that the representations constructed in §4.1 are indeed the irreducible representations of the symmetric group is given in §4.2; the proof of Frobenius’ formula, as we

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Lie Groupserentiable manifolds and maps between them, but no more; in particular, we do not mention vector fields, differential forms, Riemannian metrics, or any other tensors. Section 7.3, which discusses maps of Lie groups that are covering space maps of the underlying manifolds, may be skimmed and referred

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Lie Algebras and Lie Groupse logically necessary for what follows; §8.1 is essential. We use here a little more manifold theory: specifically, the differential of a map of manifolds is used in a fundamental way in §8.1, the notion of the tangent vector to an arc in a manifold is used in §8.2 and §8.3, and the notion of a vect

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Lie Algebras in Dimensions One, Two, and Threeess. We will work primarily with complex Lie algebras and Lie groups, but will mention the real case as well. Needless to say, this lecture is logically superfluous; but it is easy, fun, and serves a didactic purpose, so why not read it anyway. The analyses of both the Lie algebras and the Lie group

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Representations of sl2ℂ the analogous parts of §12 and §13, form the paradigm for the study of finite-dimensional representations of all semisimple Lie algebras and groups. §11.2 is less central; in it we show how the analysis carried out in §11.1 can be used to explicitly describe the tensor products of irreducible repre

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The General Setup: Analyzing the Structure and Representations of an Arbitrary Semisimple Lie Algebrneral semisimple Lie algebra and its representations. It is this algorithm that we will spend the remainder of Part III carrying out for the classical algebras, and the reader who finds the general setup confusing may wish to read this lecture in parallel with, for example, Lectures 15 and 16. In pa