A proof of the shuffle conjecture


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_*$ whose degree zero part is the ring of symmetric functions $ \operatorname {Sym}[X]$ over $ \mathbb{Q}(q,t)$. We then extend these operators to an action of an algebra $ \tilde {\mathbb{A}}_{q,t}$ 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)\mapsto (q^{-1},t^{-1})$.

J. Amer. Math. Soc. 31 (2018), no. 3, 661–697