Proof of the 2-part compositional shuffle conjecture

Abstract

In a recent paper [9] J. Haglund, J. Morse and M. Zabrocki advanced a refinement of the Shuffle Conjecture of Haglund et. al. [8]. They introduce the notion of “touch composition” of a Dyck path, whose parts yield the positions where the path touches the diagonal. They conjectured that the polynomial $\big\langle \nabla \mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots \mathbf{C}_{p_{k}}\,1\,\ h_{\mu _{1}}h_{\mu _{2}}\cdots h_{\mu _{l}}\big\rangle$, where $\mathbf{C}_{p_{1}}\mathbf{C}_{p_{2}}\cdots \mathbf{C}_{p_{k}}\,1$ is essentially a rescaled Hall–Littlewood polynomial and $\nabla$ is the Macdonald eigen-operator introduced in [1], enumerates by $t^{\mathrm{area}}q^{\mathrm{dinv}}$ the parking functions whose Dyck paths hit the diagonal by (p ₁, p ₂, …, p _k ) and whose diagonal word is a shuffle of l increasing words of lengths μ ₁, μ ₂, …, μ _l . In this paper we prove the case l = 2 of this conjecture.

In honor of Nolan Wallach

The first author was supported by NSF Grant DMS10-68883, the second author was supported by NSF of China Grant 11171231, and the third author was supported by NSERC.