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\Large{Oral exam syllabus}

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\normalsize{Michael Weingart}
   
 
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\large{Major topic: Representation Theory}
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$\bullet$ Essentials of representation theory for finite groups: characters, orthogonality relations, Schur's lemma, complete reducibility

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$\bullet$ Lie groups: definitions and examples.  The exponential map.

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$\bullet$ Explicit constructions of roots, weights, fundamental weights, Weyl groups for classical groups

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$\bullet$ Theorem of the highest weight

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$\bullet$ Construction of fundamental representations; Cartan products; weights of irreducible representations

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$\bullet$ Explicit constructions of irreducible representations of classical groups

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$\bullet$ Representations of $S_k$ and $Gl_n$ (duality, parametrization of representations by Young diagrams)

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$\bullet$ Clifford algebras and spin representations

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$\bullet$ Representations of complex simple groups; representation
rings and characters; homogeneous spaces (Borel-Weil theorem,
fundamentals of Bruhat decomposition and flag varieties)

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$\bullet$ Weyl character and dimension formulas; Freudenthal, Kostant, and Steinberg multiplicity formulas; applications and proofs (at least of WCF).

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$\bullet$ Complete reducibility of representations of semisimple algebras (algebraic and analytic proofs)

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%$\bullet$ Real Lie algebras and groups (Knapp's classification?)

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\large{Minor topic: Lie algebras}
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$\bullet$ Definitions; solvable and nilpotent algebras; theorems of Engel and Lie
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$\bullet$ Cartan's criterion for solvability; Killing form properties, and criterion for semisimplicity
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$\bullet$ Root systems from an axiomatic point of view; bases, Weyl
chambers, Weyl group

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$\bullet$ Classification of Dynkin diagrams and irreducible root
systems; affine root systems and generalized Cartan matrices
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$\bullet$ Isomorphism and conjugacy theorems: Cartan and Engel subalgebras, conjugacy of Cartan and Borel subalgebras
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$\bullet$ Construction of U(g). PBW: statement, proof, and consequences
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$\bullet$ Serre's (existence) theorem 
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$\bullet$ Representations from Humphreys' (purely algebraic) viewpoint
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References:

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R. Goodman and N. Wallach, \it{Representations and Invariants of the Classical Groups}

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W. Fulton and J. Harris, \it{Representation Theory} 

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J. Humphreys, \it{An Introduction to Lie Algebras and Representation Theory}

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W.Fulton, \it{Young Tableaux}
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