On the Schur expansion of Hall-Littlewood and related polynomials via Yamanouchi words

• Main Author: Roberts, Austin
• Format: Report
• Language: unknown
• Published: arXiv 2014
• Subjects: Combinatorics math.CO; FOS Mathematics; sami
• Online Access: https://dx.doi.org/10.48550/arxiv.1404.1036; https://arxiv.org/abs/1404.1036

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for 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 $δ\subset {\mathbb Z} \times {\mathbb Z}$, written as $\widetilde H_δ(X;q,t)$ and $\widetilde H_δ(X;0,t)$, respectively. We then give an explicit Schur expansion of $\widetilde H_δ(X;0,t)$ 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 $\widetilde H_δ(X;0,t)$ and similarly describe its Schur expansion. We then analyze $R_{γ,δ}(X)$ to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of $\widetilde H_δ(X;q,t)$. To gain these results, we use a construction from the 2007 work of Sami Assaf to 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.