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We introduce the family of Theta operators $\Theta_f$ indexed by symmetricfunctions $f$ that allow us to conjecture a compositional refinement of theDelta conjecture of Haglund, Remmel and Wilson for $\Delta_{e_{n-k-1}}'e_n$. Weshow that the $4$-variable Catalan theorem of Zabrocki is precisely theSchr\"{o}der case of our compositional Delta conjecture, and we show how torelate this conjecture to the Dyck path algebra introduced by Carlsson andMellit, extending one of their results. Again using the Theta operators, weconjecture 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 theshuffle 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 $k=0$ ofour generalized Delta square conjecture for$\frac{[n-k]_t}{[n]_t}\Delta_{h_m}\Delta_{e_{n-k}}\omega(p_n)$, extending thesquare theorem of Sergel to a generalized square theorem for$\Delta_{h_m}\nabla \omega(p_n)$. Still the Theta operators will provide aconjectural formula for the Frobenius characteristic of super-diagonalcoinvariants with two sets of Grassmanian variables, extending the one ofZabrocki for the case with one set of such variables. We propose acombinatorial interpretation of this last formula at $q=1$, leaving open theproblem of finding a dinv statistic that gives the whole symmetric function.

Theta operators, refined Delta conjectures, and coinvariants