Theta operators, refined Delta conjectures, and coinvariants

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