On the Schur Expansion of Hall-Littlewood and Related Polynomials via Yamanouchi Words

• Published in: The Electronic Journal of Combinatorics
• Main Author: Roberts, Austin
• Format: Article in Journal/Newspaper
• Language: English
• Published: The Electronic Journal of Combinatorics 2017
• Subjects: Hall-Littlewood polynomials; Dual equivalence; Schur functions; Symmetric functions; Macdonald polynomials; 05E05; 05E10; sami
• Online Access: https://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p57; https://doi.org/10.37236/6732

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 $\delta \subset {\mathbb Z} \times {\mathbb Z}$, written as $\widetilde H_{\delta}(X;q,t)$ and $\widetilde H_{\delta}(X;0,t)$, respectively. We then give an explicit Schur expansion of $\widetilde H_{\delta}(X;0,t)$ 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 $R_{\gamma,\delta}(X)$ as a refinement of $\widetilde H_{\delta}(X;0,t)$ and similarly describe its Schur expansion. We then analyze $R_{\gamma,\delta}(X)$ to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of $\widetilde H_{\delta}(X;q,t)$. 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 $\mathcal{H}_\delta$. In the case where a subgraph of $\mathcal{H}_\delta$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.