Theta operators, refined Delta conjectures, and coinvariants

Abstract

We introduce the family of Theta operators Θf indexed by symmetric functions f that allow us to conjecture a compositional refinement of the Delta conjecture of Haglund, Remmel and Wilson [23] for Δen−k−1′en. We show that the 4-variable Catalan theorem of Zabrocki [31] is precisely the Schröder case of our compositional Delta conjecture, and we show how to relate this conjecture to the Dyck path algebra introduced by Carlsson and Mellit in [6], extending one of their results.Again using the Theta operators, we conjecture a touching refinement of the generalized Delta conjecture for ΔhmΔen−k−1′en, and prove the case k=0, which was also conjectured in [23], extending the shuffle theorem of Carlsson and Mellit to a generalized shuffle theorem for Δhm∇en. Moreover we show how this implies the case k=0 of our generalized Delta square conjecture for [n−k]t[n]tΔhmΔen−kω(pn), extending the square theorem of Sergel [27] to a generalized square theorem for Δhm∇ω(pn).Still the Theta operators will provide a conjectural formula for the Frobenius characteristic of super-diagonal coinvariants with two sets of Grassmannian variables, extending the one of Zabrocki in [30] for the case with one set of such variables. We propose a combinatorial interpretation of this last formula at q=1, leaving open the problem of finding a dinv statistic that gives the whole symmetric function.