Dual equivalence graphs, ribbon tableaux and Macdonald polynomials

We make a systematic study of a new combinatorial construction called a dual equivalence graph. Motivated by the dual equivalence relation on standard Young tableaux introduced by Haiman, we axiomatize such constructions and prove that the generating functions of these graphs are Schur positive. We construct a graph on k-ribbon tableaux which we conjecture to be a dual equivalence graph, and we prove the conjecture for k ≤ 3. This implies the Schur positivity of the k-ribbon tableaux generating functions, e G (k) μ (x; q), introduced by Lascoux, Leclerc and Thibon. From Haglund’s monomial expansion for Macdonald polynomials, this has the further consequence of a combinatorial Schur expansion of the transformed Macdonald polynomials e Hμ(x; q, t) when μ is a partition with at most 3 columns.