# 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 $V_*$ 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 $(q,t)\mapsto (q^{-1},t^{-1})$.

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