# A proof of the shuffle conjecture

Erik Carlsson, Anton Mellit
### 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})$.

Publication

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