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 $δ ⊂ \mathbb{Z} \times \mathbb{Z}$, written as $\widetilde H_δ (X;q,t)$ and $\widetilde P_δ (X;t)$, respectively. We then give an explicit Schur expansion of $\widetilde P_δ (X;t)$ 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 $R_γ ,δ (X)$ as a refinement of $\widetilde P_δ$ and similarly describe its Schur expansion. We then analysize $R_γ ,δ (X)$ to determine the leading term of its Schur expansion. To gain these results, we associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_δ$ . In the case where a subgraph of $\mathcal{H}_δ$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.
Highlights
Adriano Garsia posed the question, when can the modified Hall-Littlewood polynomials Pμ(X; t) be expanded into the Schur functions as a particular sum over the Yamanouchi words, and is there a way to fix the expansion when it is not? The results of this paper are in direct response to Garsia’s question
The Macdonald polynomials were introduced in Macdonald (1988) and are often defined as the set of q, t-symmetric functions satisfying certain orthogonality and triangularity conditions
Expanding Hall-Littlewood polynomials into Schur functions can be achieved via the charge statistic, as found in Lascoux and Schutzenberger (1978), though we will present a new expansion in this paper
Summary
Adriano Garsia posed the question, when can the modified Hall-Littlewood polynomials Pμ(X; t) be expanded into the Schur functions as a particular sum over the Yamanouchi words, and is there a way to. Expanding Hall-Littlewood polynomials into Schur functions can be achieved via the charge statistic, as found in Lascoux and Schutzenberger (1978), though we will present a new expansion in this paper. We use the statistics defined in Haglund et al (2005), to generalize the definition for the modified Macdonald polynomials Hμ(X; q, t) and the modified Hall-Littlewood polynomials Pμ(X; t) to any diagram δ ⊂ Z × Z, giving the functions Hδ(X; q, t) and Pδ(X; t). The theory was further advanced by the author in Roberts (2013), from which we will derive the definition of dual equivalence graph used in this paper. The main contribution of this paper to the theory of dual equivalence graphs can be stated in the following theorem.
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