## A proof of the shuffle conjecture

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Erik Carlsson and Anton Mellit
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## 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 $\tilde {\mathbb A}$ 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 $(q,t)\mapsto (q^{-1},t^{-1})$.## References

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## Additional Information

**Erik Carlsson**- Affiliation: International Centre for Theoretical Physics, Str. Costiera, 11, 34151 Trieste, Italy
- Address at time of publication: Department of Mathematics, University of California, Davis, 1 Shields Ave., Davis, California 95616
- MR Author ID: 793205
- Email: ecarlsson@math.ucdavis.edu
**Anton Mellit**- Affiliation: International Centre for Theoretical Physics, Str. Costiera, 11, 34151 Trieste, Italy
- Address at time of publication: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
- MR Author ID: 739689
- Email: anton.mellit@univie.ac.at
- Received by editor(s): March 29, 2016
- Received by editor(s) in revised form: August 29, 2017, and October 11, 2017
- Published electronically: November 30, 2017
- © Copyright 2017 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**31**(2018), 661-697 - MSC (2010): Primary 05E10; Secondary 05E05, 05A30, 33D52
- DOI: https://doi.org/10.1090/jams/893
- MathSciNet review: 3787405