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.