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A proof of the shuffle conjecture


Authors: Erik Carlsson and Anton Mellit
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
Published electronically: November 30, 2017
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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})$.


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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
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
Email: anton.mellit@univie.ac.at

DOI: https://doi.org/10.1090/jams/893
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
Article copyright: © Copyright 2017 American Mathematical Society

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