Theta operators at $t=1$, Macdonald cumulants, and LLT positivity

Source abstract

We study the Theta operators of D'Adderio-Iraci-Vanden Wyngaerd at $t=1$ and show that the power-sum-indexed operators $Θ_{\mathsf p_k}|_{t=1}$ agree with a commuting family of derivations when restricted to symmetric functions of positive degree. Writing $\widetilde{h}_a$ for the modified Macdonald function indexed by the single row $(a)$ we establish that $Θ_{\mathsf p_μ}\widetilde{h}_a|_{t=1}$ is, up to a normalization, the single-row Macdonald cumulant of Dolęga. We use these results to show that $Θ_{\mathsf p_μ}\widetilde{h}_a|_{t=1}$ and $Θ_{\mathsf e_λ}\widetilde{h}_a|_{t=1}$ are both sums of vertical-strip LLT polynomials indexed by certain plane trees. The former further shows that single-row Macdonald cumulants are LLT-positive, thereby yielding a stronger form of the higher-order Macdonald positivity conjecture of Dolęga when all shapes are single rows.