Type A partially-symmetric Macdonald polynomials

Abstract

We construct type A partially-symmetric Macdonald polynomials P (λ∣γ) , where λ∈ℤ ≥0 n-k is a partition and γ∈ℤ ≥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 (λ∣γ) for j>n-k and e 1 [x 1 ,⋯,x n-k ]P (λ∣γ) , along with substantial combinatorial simplification of the e 1 multiplication. The P (λ∣γ) are the same as the m-symmetric Macdonald polynomials defined by Lapointe in [9] up to a change of variables.