Theta operators, refined Delta conjectures, and coinvariants

Abstract

We introduce the family of Theta operators $\Theta_f$ indexed by symmetric functions $f$ that allow us to conjecture a compositional refinement of the Delta conjecture of Haglund, Remmel and Wilson for $\Delta_{e_{n-k-1}}'e_n$. We show that the $4$-variable Catalan theorem of Zabrocki is precisely the Schr\"{o}der case of our compositional Delta conjecture, and we show how to relate this conjecture to the Dyck path algebra introduced by Carlsson and Mellit, extending one of their results. Again using the Theta operators, we conjecture a touching refinement of the generalized Delta conjecture for $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$, and prove the case $k=0$, extending the shuffle theorem of Carlsson and Mellit to a generalized shuffle theorem for $\Delta_{h_m}\nabla e_n$. Moreover we show how this implies the case $k=0$ of our generalized Delta square conjecture for $\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)$, extending the square theorem of Sergel to a generalized square theorem for $\Delta_{h_m}\nabla \omega(p_n)$. Still the Theta operators will provide a conjectural formula for the Frobenius characteristic of super-diagonal coinvariants with two sets of Grassmanian variables, extending the one of Zabrocki 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.