Skip to Main content Skip to Navigation

# Hall-Littlewood Polynomials in terms of Yamanouchi words

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.
Keywords :
Document type :
Conference papers
Domain :
Complete list of metadata

Cited literature [3 references]

https://hal.inria.fr/hal-01207555
Contributor : Coordination Episciences Iam <>
Submitted on : Thursday, October 1, 2015 - 9:28:19 AM
Last modification on : Tuesday, March 7, 2017 - 3:25:50 PM
Long-term archiving on: : Saturday, January 2, 2016 - 10:38:55 AM

### File

dmAT0163.pdf
Publisher files allowed on an open archive

### Identifiers

• HAL Id : hal-01207555, version 1

### Citation

Austin Roberts. Hall-Littlewood Polynomials in terms of Yamanouchi words. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.727-740. ⟨hal-01207555⟩

Record views

Files downloads