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Annals of Combinatorics

, Volume 23, Issue 2, pp 317–333 | Cite as

A Proof of the Delta Conjecture When \(\varvec{q=0}\)

  • Adriano Garsia
  • Jim Haglund
  • Jeffrey B. Remmel
  • Meesue YooEmail author
Article
  • 20 Downloads

Abstract

In Haglund et al. (Trans. Amer. Math. Soc. 370(6):4029–4057, 2018), Haglund, Remmel and Wilson introduce a conjecture which gives a combinatorial prediction for the result of applying a certain operator to an elementary symmetric function. This operator, defined in terms of its action on the modified Macdonald basis, has played a role in work of Garsia and Haiman on diagonal harmonics, the Hilbert scheme, and Macdonald polynomials (Garsia and Haiman in J. Algebraic Combin. 5:191–244, 1996; Haiman in Invent. Math. 149:371–407, 2002). The Delta Conjecture involves two parameters qt; in this article we give the first proof that the Delta Conjecture is true when \(q=0\) or \(t=0\).

Keywords

Delta Conjecture Macdonald polynomials Cauchy kernel method 

Mathematics Subject Classification

Primary: 05E05 Secondary: 05E10 

Notes

Acknowledgements

A. Garsia was supported by NSF Grant DMS-1700233. J. Haglund was supported by NSF grant DMS-1600670. M. Yoo was supported by NRF Grants 2016R1A5A1008055 and 2017R1C1B2005653.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Adriano Garsia
    • 1
  • Jim Haglund
    • 2
  • Jeffrey B. Remmel
    • 1
  • Meesue Yoo
    • 3
    Email author
  1. 1.Department of MathematicsUniversity of CaliforniaSan DiegoUSA
  2. 2.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  3. 3.Department of MathematicsDankook UniversityCheonanRepublic of Korea

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