Type A Partially-Symmetric Macdonald Polynomials

Abstract

We construct type A partially-symmetric Macdonald polynomials $P_{(\lambda \mid \gamma)}$, where $\lambda \in \mathbb{Z}_{\geq 0}^{n-k}$ is a partition and $\gamma \in \mathbb{Z}_{\geq 0}^k$ 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 $e_1[x_1, \dotsc, x_{n-k}] P_{(\lambda \mid \gamma)}$, 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.