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\begin{document}


\title{Topics for oral qualifying exam for Waldeck Schutzer, Spring 2001 }

\maketitle

\section{Special Topics}


\subsection{The Littlewood-Richardson Rule}

The product of two Schur functions and Skew Schur functions , Tableaux with
lattice permutation property, statement of the Rule, the Robinson bijection
and the proof of the Rule.


\subsection{The Weyl Character Formula}

Statement of the WCF and proof for the special case of \( \textrm{GL}_{n} \).


\section{Lie Algebras }


\subsection{Basic concepts }

The notion of Lie algebra. Linear Lie algebras. Lie algebras of derivations.
Abstract Lie algebras. Ideals. Homomorphisms and representations. Automorphisms.
Solvability. Nilpotency. Engel's theorem.


\subsection{Semisimple Lie algebras }

Lie's theorem. Jordan-Chevalley decomposition. Cartan's criterion. Criterion
for semisimplicity. Simple ideals. Inner derivations. Abstract Jordan decomposition.
Modules. Casimir element of a representation. Weyl's theorem. Preservation of
Jordan decomposition. Weights and maximal vectors. Classification of irreducible
\( \mathfrak {sl}_{n}\mathbb {C} \)-modules. Maximal toral subalgebras and
roots. Centralizer of \( H \). Orthogonality, Integrality and Rationality properties.


\subsection{Root Systems}

Reflections in a euclidean plane. Root systems. Pairs of roots. Bases and Weyl
chambers. Simple roots. The Weyl group. Irreducible root systems. Cartan matrix.
Coxeter graphs and Dynkin diagrams. Irreducible components. Classification theorem.
Abstract theory of weights.


\subsection{Existence Theorem}

Universal enveloping algebras. Tensor and symmetric algebras. Construction of
\( \mathfrak {U}(L) \). PBW theorem and its consequences. Free Lie algebras.
Generators and relations. Relations satisfied by \( L \). Serre's theorem.
Existence and uniqueness theorems. The simple Lie algebras. Criterion for semisimplicity.
The classical algebras.


\section{Symmetric Functions }


\subsection{Partitions }

The notion of partition, partition diagrams, conjugate of a partition, skew
diagrams and tableaux, addition and multiplication of partitions, the natural
ordering, raising operators, the hook-length and the content.


\subsection{The ring of symmetric functions }

The ring \( \Lambda  \) of symmetric functions, monomial, elementary, complete
and forgotten symmetric functions, the generating functions \( E(t) \), \( H(t), \)
and \( P(t) \), the involution \( \omega  \), power sums and the ring of symmetric
functions \( \Lambda _{\mathbb {Q}} \) over the rationals, a few specializations.


\subsection{Schur functions }

Definition of Schur polynomial, a determinantal identity, Schur functions


\subsection{Orthogonality}

Cauchy formula, an \( \omega  \)-invariant inner product in \( \Lambda  \),
orthogonal and orthonormal bases.


\subsection{Skew Schur functions }

Definition of Skew Schur functions, A determinantal identity, the Kostka numbers.


\subsection{Transition matrices }

The notion of transition matrix and the transition matrices corresponding to
the bases \( m,e,h,f \) and \( s \)


\subsection{The characters of symmetric groups }

The representation ring \( R^{n} \) of the symmetric group \( S_{n} \), the
ring \( R=\oplus _{n\geq 0}R^{n} \), the characteristic map from \( R \) to
\( \Lambda  \). The irreducible characters and the character table of \( S_{n} \).
Frobenius formula and Young's rule. The internal product.

\begin{thebibliography}{}
\bibitem{1}William Fulton \& Joe Harris, \emph{Representation Theory - A First Course},
GTM 129, Springer-Verlag, 1991.
\bibitem{2}James E. Humphreys, \emph{Introduction to Lie Algebras and Representation Theor}y,
Third Printing, GTM 9, Springer-Verlag, 1980.
\bibitem{3}Nathan Jacobson, \emph{Lie Algebras}, Dover Publications, Inc, 1979.
\bibitem{4}Ian G. MacDonald, \emph{Symmetric Functions and Hall Polynomials}, Second Edition,
Reprint, Oxford Science Publications, 1998.
\end{thebibliography}
\end{document}
