A proof of the q,t-Schröder conjecture

Abstract

We prove a recent conjecture of Egge, Haglund, Killpatrick, and Kremer (2003), which gives a combinatorial formula for the coefficients of a hook shape in the Schur function expansion of the symmetric function ∇ en, which Haiman (2002) has shown to have a representation-theoretic interpretation. More precisely, we show that 〈∇en, en−dhd〉 can be expressed as ∑ qareatbounce, where the sum is over all Schröder lattice paths, and area and bounce are simple statistics on these paths. For d=0, this reduces to Garsia and Haglund's formula for the q,t-Catalan sequence (2001). Our results build on symmetric function identities for sums of generalized Pieri coefficients and Macdonald polynomials due to Bergeron, Garsia, Haiman, and Tesler (1999) and Garsia and Haglund (2002). We also derive several transformation identities for sums of rational functions occurring in the theory of Macdonald polynomials and diagonal harmonics, and apply these to obtain a combinatorial formula for 〈∇en, hn−dhd〉. We discuss how our formulas for 〈∇en, en−dhd〉 and 〈∇en, hn−dhd〉 prove two special cases of a recent conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov.