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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sl4? and sln? specific Lie algebras carried out in this Part. We start in §15.1 by describing the Cartan subalgebra, roots, root spaces, etc., for.in general. We then give in §15.2 a detailed account of the representations of., which generalizes directly to.in particular, we deduce the existence part of Theorem

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Symplectic Lie Algebrasbe in general the structure of a symplectic Lie algebra (that is, give a Cartan subalgebra, find the roots, describe the Killing form, and so on). We will then work out in some detail the representations of the specific algebra .. As in the case of the corresponding analysis of the special linear Li

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Weyl’s Constructiong these to the standard representation.of .. While it may be easiest to read this material while the definitions of the Young symmetrizers are still fresh in the mind, the construction will not be used again until §15, so that this lecture can be deferred until then.

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Representations of sl2? §11.1 and §11.2 are completely elementary (we do use the notion of symmetric powers of a vector space, but in a non-threatening way). §11.3 involves a fair amount of classical projective geometry, and can be skimmed or skipped by those not already familiar with the relevant basic notions from algebraic geometry.

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Representations of,and,t we know about the symmetric group; this should be completely straightforward given just the basic ideas of the preceding lecture. In the latter case we start essentially from scratch. The two sections can be read (or not) independently; neither is logically necessary for the remainder of the book.

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Lie Groupsy 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 back to as needed, though working through it will help promote familiarity with basic examples of Lie groups.

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Representations ofsl3?, Part II: Mainly Lots of Examplesne its multiplicities. The latter two sections correspond to §11.2 and §11.3 in the lecture on .. In particular, §13.4, like §11.3, involves some projective algebraic geometry and may be skipped by those to whom this is unfamiliar.