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Conjecture n! et généralisations

Abstract : This thesis deals with the question of algebraic combinatorics called the n! conjecture.

More precisely, we study the structure of some spaces M_mu indexed by the partitions mu of the integer n. Each space M_mu is a cone spanned by all partial derivatives of a polynomial Delta_mu which generalizes the Vandermonde determinant. Our work is focused on the n! conjecture, stated in 1991 by A. Garsia and M. Haiman, and recently proved by Haiman. It is motivated by the interpretation of some Macdonald polynomials with multiplicities of irreducible representations of the S}_n-module M_mu.

We first search explicit monomial bases for the spaces M_mu. This approach is closely related to the study of the vanishing ideal of Delta_mu and leads us to introduce some very useful derivative operators called shift operators. We obtain an explicit monomial basis and a description of the vanishing ideal in the case of the hook partitions, and for the subspace in one alphabet M_mu(X), for any partition mu.

The shift operators are also crucial in the introduction and the study of generalizations of the n! conjecture. In the case of punctured diagrams (recursive approach of the n! conjecture), the description of an explicit basis of the subspace in one alphabet gives us a specialization of the central four term recurrence. In the case of diagrams with many holes, the introduction of sums of cones leads to a conjectural generalization of the n! conjecture. An upper bound and the structure of the subspace in one alphabet supports this conjecture.
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Contributor : Jean-Christophe Aval <>
Submitted on : Monday, November 5, 2007 - 11:03:09 AM
Last modification on : Friday, March 13, 2020 - 10:48:03 AM
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Jean-Christophe Aval. Conjecture n! et généralisations. Mathématiques [math]. Université Sciences et Technologies - Bordeaux I, 2001. Français. ⟨tel-00185056⟩

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