A proof of the shuffle conjecture

Abstract

We present a proof of the compositional shuffle conjecture by Haglund, Morse and Zabrocki [Canad. J. Math., 64 (2012), 822–844], which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra by Haglund, Haiman, Loehr, Remmel, and Ulyanov [Duke Math. J., 126 (2005), 195–232]. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector spaceV∗V_*whose degree zero part is the ring of symmetric functionsSym⁡[X]\operatorname {Sym}[X]overQ(q,t)\mathbb {Q}(q,t). We then extend these operators to an action of an algebraA~\tilde {\mathbb A}acting on this space, and interpret the right generalization of the∇\nablausing an involution of the algebra which is antilinear with respect to the conjugation(q,t)↦(q−1,t−1)(q,t)\mapsto (q^{-1},t^{-1}).