A proof of the shuffle conjecture

Authors:
Erik Carlsson and Anton Mellit

Journal:
J. Amer. Math. Soc.

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 whose degree zero part is the ring of symmetric functions over . We then extend these operators to an action of an algebra acting on this space, and interpret the right generalization of the using an involution of the algebra which is antilinear with respect to the conjugation .

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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