About my thesis

Get my thesis, 1996; it describes new results on Schubert polynomials, new programs in algebraic combinatorics and some experiences in distributed computations applied to combinatorics.

(ACE 1.0 1995, ACE 2.0 1996, ACE 3.0 1998 );

I started my thesis in October 1994, under the responsability of J.-Y. Thibon, professor at the Université de Marne-la-Vallée. In fact, I have mainly collaborated with A. Lascoux, directeur de recherche at CNRS, in order to develop programs related to algebraic combinatorics and also to use Schubert polynomials to compute in the ring of polynomials in several variables.

Schubert polynomials are polynomials in several variables, introduced in 1982, by A. Lascoux and M.P. Schützenberger for some question in algebraic combinatorics. However, their interest is not limited to algebraic combinatorics. These polynomials form a linear basis of the space of polynomials in several variables, well-adapted to different kinds of computations. They are for instance, in the core of the computer algebra system SYMMETRICA (developed by the University of Bayreuth, in collaboration with other universities), devoted to computations on symmetric functions and representation theory of classical groups. They also provide a generalization of the famous Newton interpolation formula, in the case of several variables.

Combinatorics problems linked to these polynomials can be treated independently from their various geometrical applications. However, these applications are not yet available in SYMMETRICA. It then would be useful to write in the computer algebra system MAPLE, a specialized program of the same type as the package SCHUBERT by S. Katz and S.A. Stromme (which solely uses symmetric functions by using the SF package by J. Stembridge). The package SCHUBERT solely offers to work on grassmannians, while Schubert polynomials may be used to compute in flag varieties.

Thus, the aim was to write such a program using all possibilities of Schubert polynomials and to provide a way to compute with flag varieties. It was also planned to incorporate Grothendieck polynomials (recently introduced by A. Lascoux and M.P. Schützenberger). These polynomials, which provide more accurate computations on flag varieties, are not yet available in any system, while their rich combinatorics properties have not been explored far away. Finding the structure coefficients of the ring of polynomials in both Schubert and Grothendieck bases is not yet solved. Specializations of these polynomials are also very interesting and closely linked to well-known integers combinatorics families. The result is ACE 3.0.

We had also in mind to use parallel and distributed computations in algebraic combinatorics.

Distributed computations:

Performance analysis with PGPVM2, 1997;
Applied to algebraic computations using HUB, 1997.

Related publications


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