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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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A proof of the shuffle conjecture
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by Erik Carlsson and Anton Mellit HTML | PDF
J. Amer. Math. Soc. 31 (2018), 661-697 Request permission

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
  • 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