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

Austin Roberts¹

¹1QBit Information Technologies & Highline College

The electronic journal of combinatorics, Tome 24 (2017) no. 1

Résumé

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.

Détail

DOI : 10.37236/6732

Classification : 05E05, 14N15, 33D52
Mots-clés : Hall-Littlewood polynomials, dual equivalence, Schur functions, symmetric functions, Macdonald polynomials