Schedules and the Delta Conjecture

Abstract

In a recent preprint, Carlsson and Oblomkov (Affine Schubert calculus and double coinvariants. arXiv preprint 1801.09033, 2018) obtain a long sought-after monomial basis for the ring \(\mathrm{D}\!\mathrm{R}_n\) of diagonal coinvariants. Their basis is closely related to the “schedules” formula for the Hilbert series of \(\mathrm{D}\!\mathrm{R}_n\) which was conjectured by the first author and Loehr (Discete Math 298(1–3):189–204, 2005) and first proved by Carlsson and Mellit (A proof of the shuffle conjecture. J Amer Math Soc 31(3):661–697, 2018), as a consequence of their proof of the famous Shuffle Conjecture. In this article, we obtain a schedules formula for the combinatorial side of the Delta Conjecture, a conjecture introduced by the Haglund et al. (Trans Am Math Soc 370(6):4029–4057, 2018), which contains the Shuffle Theorem as a special case. Motivated by the Carlsson–Oblomkov basis for \(\mathrm{D}\!\mathrm{R}_n\) and our Delta schedules formula, we introduce a (conjectural) basis for the super-diagonal coinvariant ring \(\mathrm{S}\!\mathrm{D}\!\mathrm{R}_n\), an \(S_n\)-module generalizing \(\mathrm{D}\!\mathrm{R}_n\) introduced recently by Zabrocki (a module for the Delta conjecture. arXiv preprint 1902.08966, 2019), which conjecturally corresponds to the Delta Conjecture.

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Acknowledgements

The authors gratefully acknowledge NSF support for this work; the first author by Grant DMS-1600670 and the second by grant DMS-1603681. The authors would also like to thank Mike Zabrocki for suggesting the method used in Sect. 4.2 (private correspondence), as well as Alessandro Iraci and Anna Vanden Wyngaerd for helping to improve the exposition.

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Haglund, J., Sergel, E. Schedules and the Delta Conjecture. Ann. Comb. (2020). https://doi.org/10.1007/s00026-020-00517-0

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Keywords

  • Delta conjecture
  • Parking function
  • Coinvariant ring
  • Super-space

Mathematics Subject Classification

  • 05E10
  • 05E05