Abstract
We express the integral form Macdonald polynomials as a weighted sum of Shareshian and Wachs' chromatic quasisymmetric functions of certain graphs. Then we use known expansions of these chromatic quasisymmetric functions into Schur and power sum symmetric functions to provide Schur and power sum formulas for the integral form Macdonald polynomials. Since the (integral form) Jack polynomials are a specialization of integral form Macdonald polynomials, we obtain analogous formulas for Jack polynomials as corollaries.
Highlights
The Macdonald polynomials are a basis {Pμ(x; q, t) : partitions μ} for the ring of symmetric functions which are defined by certain triangularity relations [Mac95]
This basis has the additional property that it reduces to classical bases, such as the Schur functions and the monomial symmetric functions, after certain specializations of q and t
The Jack polynomials are another basis for the ring of symmetric functions which are obtained by taking a certain limit of integral form Macdonald polynomials
Summary
The Macdonald polynomials are a basis {Pμ(x; q, t) : partitions μ} for the ring of symmetric functions which are defined by certain triangularity relations [Mac95]. Shareshian and Wachs prove that XG(x; t) is symmetric for a certain class of graphs G; in particular, the graphs in this paper will all have symmetric XG(x; t) These XG(x; t) have known formulas for their Schur [SW16] and power sum [Ath15] expansions. We use this formula to obtain Schur and power sum expansions of integral form Macdonald polynomials. The Jack polynomials are another basis for the ring of symmetric functions which are obtained by taking a certain limit of integral form Macdonald polynomials.
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