Abstract
We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.
Highlights
While working towards a proof of the Schur positivity of Macdonald polynomials, Garsia and Haiman discovered the module of diagonal harmonics, an Sn-module that captures many of the properties of Macdonald polynomials
In [Hai02], Haiman proved that the Frobenius characteristic of the module of diagonal harmonics could be written as ∇en or ∆en en for certain symmetric function operators ∇ and ∆f which are eigenoperators of Macdonald polynomials
Haiman, Loehr, Ulyanov, and the first two authors proposed a connection between ∇en and parking functions which has come to be known as the Shuffle Conjecture [HHL+05]
Summary
While working towards a proof of the Schur positivity of Macdonald polynomials, Garsia and Haiman discovered the module of diagonal harmonics, an Sn-module that captures many of the properties of Macdonald polynomials. In [Hai02], Haiman proved that the Frobenius characteristic of the module of diagonal harmonics could be written as ∇en or ∆en en for certain symmetric function operators ∇ and ∆f which are eigenoperators of Macdonald polynomials. Haiman, Loehr, Ulyanov, and the first two authors proposed a connection between ∇en and parking functions which has come to be known as the Shuffle Conjecture [HHL+05]. The goal of this abstract is to state and support two versions of a generalization of the Shuffle Conjecture in which ∆en en is replaced by ∆ek en for an. We provide the necessary background and state our conjecture
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