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 space V ∗ V_* whose degree zero part is the ring of symmetric functions Sym ⁡ [ X ] \operatorname {Sym}[X] over Q ( q , t ) \mathbb {Q}(q,t) . We then extend these operators to an action of an algebra A ~ \tilde {\mathbb A} acting on this space, and interpret the right generalization of the ∇ \nabla using 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}) .