Aspects combinatoires des polynômes de MacDonald

La théorie sur les polynômes de Macdonald a fait l'objet d'une quantité importante de recherches au cours des dernières années. Définis originellement par Macdonald comme une généralisation de quelques-unes des bases les plus importantes de l'anneau des fonctions symétriques, ces polynômes ont des applications dans des domaines tels que la théorie des représentations des groupes quantiques et physique des particules. Ce travail présente quelques-uns des résultats combinatoires les plus importants entourant ces polynômes, en mettant particulièrement l'accent sur la formule combinatoire prouvée récemment par Haglund, Haiman et Loehr pour les polynômes de Macdonald. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Polynômes de Macdonald, Fonctions symétriques, Combinatoire algébrique, Combinatoire énumérative, Théorie des représentations, Géométrie algébrique

The combinatorics of modified Macdonald polynomials

This paper is concerned with finding a combinatorial Schur expansion of the modified Macdonald polynomials. We will use the Robinson–Schensted–Knuth (RSK) algorithm to obtain the result for a limited set of partition shapes, and we will use Foata’s bijection to extend the result to conjugate shapes. We will explore possibilities for modifying the RSK algorithm in such a way that it could be applied to obtain the result for more general shapes

Affine dual equivalence and k-Schur functions

The k-Schur functions were first introduced by Lapointe, Lascoux and Morse (2003) in the hopes of refining the expansion of Macdonald polynomials into Schur functions. Recently, an alternative definition for k-Schur functions was given by Lam, Lapointe, Morse, and Shimozono (2010) as the weighted generating function of starred strong tableaux which correspond with labeled saturated chains in the Bruhat order on the affine symmetric group modulo the symmetric group. This definition has been shown to correspond to the Schubert basis for the affine Grassmannian of type A by Lam (2008), and, at t = 1, it is equivalent to the k-tableaux characterization of Lapointe and Morse (2007). In this paper, we extend Haiman's (1992) dual equivalence relation on standard Young tableaux to all starred strong tableaux. The elementary equivalence relations can be interpreted as labeled edges in a graph which share many of the properties of Assaf's dual equivalence graphs. These graphs display much of the complexity of working with k-Schur functions and the interval structure on affine Symmetric Group modulo the Symmetric Group. We introduce the notions of flattening and squashing skew starred strong tableaux in analogy with jeu da taquin slides in order to give a method to find all isomorphism types for affine dual equivalence graphs of rank 4. Finally, we make connections between k-Schur functions and both LLT and Macdonald polynomials by comparing the graphs for these functions.Comment: 49 pages, 14 figure

Maximal chain descent orders

This paper introduces a partial order on the maximal chains of any finite bounded poset $P$ which has a CL-labeling $\lambda$. We call this the maximal chain descent order induced by $\lambda$, denoted $P_{\lambda}(2)$. As a first example, letting $P$ be the Boolean lattice and $\lambda$ its standard EL-labeling gives $P_{\lambda}(2)$ isomorphic to the weak order of type A. We discuss in depth other seemingly well-structured examples: the max-min EL-labeling of the partition lattice gives maximal chain descent order isomorphic to a partial order on certain labeled trees, and particular cases of the linear extension EL-labelings of finite distributive lattices produce maximal chain descent orders isomorphic to partial orders on standard Young tableaux. We observe that the order relations which one might expect to be the cover relations, those given by the "polygon moves" whose transitive closure defines the maximal chain descent order, are not always cover relations. Several examples illustrate this fact. Nonetheless, we characterize the EL-labelings for which every polygon move gives a cover relation, and we prove many well known EL-labelings do have the expected cover relations. One motivation for $P_{\lambda}(2)$ is that its linear extensions give all of the shellings of the order complex of $P$ whose restriction maps are defined by the descents with respect to $\lambda$. This yields strictly more shellings of $P$ than the lexicographic ones induced by $\lambda$. Thus, the maximal chain descent order $P_{\lambda}(2)$ might be thought of as encoding the structure of the set of shellings induced by $\lambda$.Comment: 48 pages, 21 figure