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\centerline{\sf  Marne-la-Vall\'ee / Paris / Rouen}

\bigskip\bigskip
\centerline{\bf \LARGE DESCRIPTION OF THE GROUP}

\vskip 20mm

\section*{General presentation}

Our group is composed of 17 researchers, among which 8 seniors and 9 graduate
students, issued from three universities: Marne-la-Vall\'ee, Paris 7 and Rouen.
The group has a very strong scientific unity, and meets every week 
in Marne-la-Vall\'ee for the Algebraic Combinatorics Seminar.


\section*{Researchers involved in the project}

\subsection*{Senior researchers}

---  Christophe {\sc Carr\'e}, Ma\^\i tre de Conf\'erences, Universit\'e de Rouen

---  Jacques {\sc D\'esarm\'enien}, Professeur, Universit\'e de Marne-la-Vall\'ee

---  G\'erard {\sc Duchamp}, Professeur, Universit\'e de Rouen

---  Daniel {\sc Krob}, Charg\'e de Recherches CNRS, Universit\'e Paris 7

---  Alain {\sc Lascoux}, Directeur de Recherches CNRS, Universit\'e Paris 7

---  Bernard {\sc Leclerc}, Charg\'e de Recherches CNRS, Universit\'e Paris 7 

---  Pierre-Andr\'e {\sc Picon}, Professeur, Universit\'e de Marne-la-Vall\'ee

---  Jean-Yves {\sc Thibon}, Professeur, Universit\'e de Marne-la-Vall\'ee


\subsection*{Graduate students}

---  Abdelhamid {\sc Abderrezzak}, Universit\'e Paris 7 

---  Philippe {\sc Andary}, Universit\'e de Rouen 

---  Renaud {\sc Eppstein}, Universit\'e de Marne-la-Vall\'ee 

---  Sungsoon {\sc Kim}, Universit\'e Paris 7

---  Eric {\sc Laugerotte}, Universit\'e de Rouen

---  Loic {\sc Le Bris}, Universit\'e de Marne-la-Vall\'ee

---  Jean-Christophe {\sc Novelli}, Universit\'e Paris 7

---  Bun-Chan-Vorac {\sc Ung}, Universit\'e de Marne-la-Vall\'ee

---  S\'ebastien {\sc Veigneau}, Universit\'e de Marne-la-Vall\'ee

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\centerline{\sf  Marne-la-Vall\'ee / Paris / Rouen}

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\centerline{\bf \LARGE RESEARCH TOPICS}

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The research  carried out by our team consists in the application
of combinatorial methods to various problems in Representation Theory,
Algebraic Geometry and Mathematical Physics. More precisely, we 
exploit the well-developed combinatorics of Young tableaux and symmetric
functions to the effect of understanding recently discovered fundamental objects
such as crystal bases of quantum groups, Schubert and Grothendieck polynomials
or Kazhdan-Lusztig polynomials. We are also interested in Lie series and
free Lie algebras, shuffle algebras and their generalizations.


\medskip
Typical examples of our recent activities are listed below.
\begin{itemize}

\item Using an explicit realization of the Hecke algebras of type $A$
by symmetrizing operators acting on the ring of polynomials, we have
described their irreducible representations and obtained a $q$-analogue
of the straightening algorithm of classical invariant theory.

\item In collaboration with I.M. Gelfand, we have developed a 
theory of noncommutative symmetric functions and discussed several
applications to Solomon's descent algebras, calculations of Lie idempotents
and Lie series, Laplace operators of universal enveloping algebras
of classical Lie algebras.

\item We have investigated the specialization of Hall-Littlewood functions 
at roots of unity, and exhibited a connection between this question
and the problem of the cyclic decomposition of tensor powers of
a finite-dimensional representation of the general linear group.
This lead us to a combinatorial description of the expansion of some particular
plethysms on the basis of Schur functions.

\item We have given a new combinatorial description of the $q$-analogues
of weight multiplicities of irreducible $GL_n$-modules in terms of the
geometry of the associated crystal graphs.

\end{itemize}

This kind of research generally leads to the elaboration of efficient algorithms
for the manipulation of the objects under study. 
Specific softwares devoted to our research topics are developed and freely distributed.
In particular, we have contributed programs to the computer algebra system
SYMMETRICA designed by the Bayreuth team under the direction of A. Kerber and
A. Kohnert.

\medskip
We have been collaborating with the following teams: Aberystwyth, Amsterdam
and Bayreuth.


%\newpage
\footnotesize
\section*{Selected recent publications}


\begin{enumerate}

\item {\sc G. Duchamp, D. Krob, A. Lascoux, B. Leclerc, T. Scharf \&
J.-Y. Thibon}, 
\it Euler-Poincar\'e characteristic and polynomial representations
of Iwahori-Hecke algebras\rm, 
Publications of the Research Institute for Mathematical
Sciences (Kyoto) 1995, to appear.

\item {\sc J. D\'esarm\'enien, G. Duchamp, D. Krob et G. M\'elan\c on},
{\it Quelques remarques sur les superalg\`ebres de Lie libres},
C. R. Acad. Sci. Paris, {\bf 318} (1994), 419-424. 

\item {\sc D. Krob \& B. Leclerc}, {\it Minor identities for quasi-determinants 
and quantum determinants}, Comm. Math. Physics {\bf 169} (1995), 1-23.

\item {\sc T. Scharf, J.-Y. Thibon \& B.G. Wybourne},
{\it Powers of the Vandermonde determinant and the quantum
Hall effect}\rm, 
Journal of Physics A {\bf 27} (1994), 4211-4219.

\item {\sc I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh \& J.-Y. Thibon}
{\it Noncommutative symmetric functions}\rm,
Advances in Mathematics
{\bf 112} (1995), 218-348.

\item {\sc G. Duchamp, D. Krob, B. Leclerc \& J.-Y. Thibon}, 
{\it D\'eformations de projecteurs de Lie}, 
C.R. Acad. Sci. Paris, {\bf 319} (1994), 909--914.

\item {\sc W. Fulton \& A. Lascoux}, {\it A Pieri formula in the
Grothendieck ring of a flag bundle}, Duke Math. J., {\bf 76} (1994), 711-729.

\item {\sc A. Lascoux, B. Leclerc \& J.-Y. Thibon},
{\it Crystal graphs and $q$-analogues of weight multiplicities
for the root system $A_n$}\rm, 
Letters in Mathematical Physics 1995, to appear.

\item {\sc C. Carr\'e \& B. Leclerc}, {\it Splitting the square
of a Schur function into its symmetric and antisymmetric parts}\rm,
Journal of Algebraic Combinatorics 1995, to appear.

\item {\sc A. Lascoux, B. Leclerc \& J.-Y. Thibon}, {\it Twisted Action 
of the Symmetric Group on the Cohomology Ring of a Flag Manifold}, 
Proceedings of the Banach Institute, Warszaw, to appear.

\item {\sc S. Veigneau}, {\it $SP$, a package for Schubert polynomials
realized with the computer algebra system MAPLE}, submitted to the
Journal of Symbolic Computation.
\end{enumerate}

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