Schedules and the Delta Conjecture

Abstract

In a recent preprint, Carlsson and Oblomkov (2018) obtain a long sought after monomial basis for the ring $\operatorname{DR}_n$ of diagonal coinvariants. Their basis is closely related to the "schedules" formula for the Hilbert series of $\operatorname{DR}_n$ which was conjectured by the first author and Loehr (2005) and first proved by Carlsson and Mellit (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 first author, Remmel and Wilson (2018) which contains the Shuffle Conjecture as a special case. Motivated by the Carlsson-Oblomkov basis for $\operatorname{DR}_n$ and our Delta schedules formula, we introduce a (conjectural) basis for the module $\operatorname{SDR}_n$ of super-diagonal coinvariants, an $S_n$ module generalizing $\operatorname{DR}_n$ introduced recently by Zabrocki (2019) which conjecturally corresponds to the Delta Conjecture.