Type A Partially-Symmetric Macdonald Polynomials

Abstract

We construct type A partially-symmetric Macdonald polynomials $P_{(\lambda \mid \gamma )}$, where $\lambda \in {\mathbb{Z}}_{\ge 0}^{n-k}$ is a partition and $\gamma \in {\mathbb{Z}}_{\ge 0}^k$ is a composition. These are polynomials which are symmetric in the first $n-k$ variables, but not necessarily in the final $k$ variables. We establish their stability and an integral form defined using Young diagram statistics. Finally, we build Pieri-type rules for degree 1 products $x_j{P}_{(\lambda \mid \gamma )}$ for $j>n-k$ and $e_1[x_1,\dots ,x_{n-k}]P_{(\lambda \mid \gamma )}$, along with substantial combinatorial simplification of the $e_1$ multiplication. The $P_{(\lambda \mid \gamma )}$ are the same as the $m$-symmetric Macdonald polynomials defined by Lapointe up to a change of variables.

arXiv comment49 pages, 2 figures