Expanding Hall-Littlewood and related polynomials as sums over Yamanouchi words

Abstract. This paper uses the theory of dual equivalence graphs to give explicit Schur expansions to several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram δ ⊂ Z × Z, written as H̃δ(X; q, t) and P̃δ(X; t), respectively. We then give an explicit Schur expansion of P̃δ(X; t) as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function Rγ,δ(X) as a refinement of P̃δ and similarly describe its Schur expansion. We then analysize Rγ,δ(X) to determine the leading term of its Schur expansion. To gain these results, we associate each Macdonald polynomial with a signed colored graph Hδ. In the case where a subgraph of Hδ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries. Résumé. Ce document utilise la théorie des graphes double équivalence pour donner expansions de Schur explicites à plusieurs familles de fonctions symétriques. Nous commençons par donner une définition combinatoire des polynômes de Macdonald modifiés et polynômes de Hall-Littlewood modifiés indexés par tout schèma δ ⊂ Z × Z, écrit H̃δ(X, q, t) et P̃δ(X, t), respectivement. Nous donnons ensuite une expansion de Schur explicite de P̃δ(X, t) comme une somme sur un sous-ensemble des mots Yamanouchi, plutôt que l’expansion en utilisant la statistique de charge donnée en 1978 par Lascoux et Schüztenberger. Nous définissons davantage la fonction symétrique Rγ,δ(X) comm