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

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 HallLittlewood polynomials indexed by any diagram δ ⊂ Z × Z, written as H̃δ(X; q, t) and H̃δ(X; 0, t), respectively. We then give an explicit Schur expansion of H̃δ(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 Rγ,δ(X) as a refinement of H̃δ(X; 0, t) and similarly describe its Schur expansion. We then analyze Rγ,δ(X) to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of H̃δ(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 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.