The nabla operator, Hall--Littlewood polynomials and LLT polynomials

Bergeron, Garsia, Haiman, and Tesler conjectured that $(-1{)}^{n-\mathrm{\ell }(\mu )}\mathrm{\nabla }{\tilde{H}}_{\mu }[X;0,t]$ and $\mathrm{\nabla }\omega {\tilde{H}}_{\mu }[X;0,t^{-1}]$ are Schur positive, and that $\mathrm{\nabla }\omega {\tilde{H}}_{\mu }[X;0,t^{-1}]-\mathrm{\nabla }\omega {\tilde{H}}_{\nu }[X;0,t^{-1}]$ is Schur positive when $\mu$ dominates $\nu$. The first statement follows from the compositional shuffle conjecture of Haglund, Morse, and Zabrocki, proved by Carlsson and Mellit. We prove the other two statements. All three statements also hold for ${\mathrm{\nabla }}^r$ with $r\ge 1$. The main new step is an identity: when $\mu$ covers $\nu$, the difference $\omega {\tilde{H}}_{\mu }[X;0,t^{-1}]-\omega {\tilde{H}}_{\nu }[X;0,t^{-1}]$ is minus a power of $t$ times one LLT polynomial. Using the Catalanimals and the Cauchy identity of Blasiak, Haiman, Morse, Pun, and Seelinger, we show that ${\mathrm{\nabla }}^r$ of this LLT polynomial is, up to sign, a positive sum of LLT polynomials. We also extend the main results to the Schiffmann algebra setting.