Hall–Littlewood Operators in the Theory of Parking Functions and Diagonal Harmonics

In a recent work, Jim Haglund, Jennifer Morse, and Mike Zabrocki proved a variety of identities involving Hall–Littlewood symmetric functions indexed by compositions. When they applied ∇ to these symmetric functions, the resulting identities and computer data led them to some truly remarkable refinements of the shuffle conjecture. We prove here the symmetric function side of a recursion which when combined with a recent parking function recursion of Angela Hicks [18] settles some special cases of the Haglund–Morse–Zabrocki conjectures. Our main result of a compositional q,t-Catalan and Schröder theorem yields, as a consequence, surprisingly simple new proofs of the original q,t-Catalan and Schröder results.