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Transactions of the American Mathematical Society

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The Delta Conjecture


Authors: J. Haglund, J. B. Remmel and A. T. Wilson
Journal: Trans. Amer. Math. Soc. 370 (2018), 4029-4057
MSC (2010): Primary 05E05
DOI: https://doi.org/10.1090/tran/7096
Published electronically: February 1, 2018
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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
Email: jhaglund@math.upenn.edu

J. B. Remmel
Affiliation: Department of Mathematics, UC San Diego, La Jolla, California 92093
Email: jremmel@math.ucsd.edu

A. T. Wilson
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Email: andwils@math.upenn.edu

DOI: https://doi.org/10.1090/tran/7096
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.
Article copyright: © Copyright 2018 American Mathematical Society

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