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Transactions of the American Mathematical Society

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

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The Delta Conjecture
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by J. Haglund, J. B. Remmel and A. T. Wilson PDF
Trans. Amer. Math. Soc. 370 (2018), 4029-4057 Request permission

Abstract:

We conjecture two combinatorial interpretations for the symmetric function $\Delta _{e_k} e_n$, where $\Delta _f$ is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture of Haglund, Haiman, Remmel, Loehr, and Ulyanov, which was proved recently by Carlsson and Mellit. We show how previous work of the third author on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.
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Additional Information
  • J. Haglund
  • Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
  • MR Author ID: 600170
  • Email: jhaglund@math.upenn.edu
  • J. B. Remmel
  • Affiliation: Department of Mathematics, UC San Diego, La Jolla, California 92093
  • MR Author ID: 146845
  • Email: jremmel@math.ucsd.edu
  • A. T. Wilson
  • Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
  • Email: andwils@math.upenn.edu
  • Received by editor(s): September 23, 2015
  • Received by editor(s) in revised form: September 14, 2016
  • Published electronically: February 1, 2018
  • Additional Notes: The first author was partially supported by NSF grant DMS-1200296.
    The third author was supported by a DoD National Defense Science and Engineering Graduate Fellowship and an NSF Mathematical Sciences Postdoctoral Research Fellowship.
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 4029-4057
  • MSC (2010): Primary 05E05
  • DOI: https://doi.org/10.1090/tran/7096
  • MathSciNet review: 3811519