The Schur positivity of $\nabla m_\mu$

Abstract

Bergeron, Garsia, Haiman and Tesler conjectured in 1999 that, for all partitions $\mu,\lambda\vdash n$, the polynomial $(-1)^{|\mu|-\ell(\mu)}\langle \nabla m_\mu, s_\lambda\rangle$ has nonnegative integer coefficients, where $\nabla$ is the Bergeron--Garsia nabla operator, which acts diagonally on the modified Macdonald basis, and $m_\mu$ is the monomial symmetric function. In this article, we prove this conjecture, and more generally that $(-1)^{|\mu|-\ell(\mu)}\langle\nabla^r m_\mu,s_\lambda\rangle\in\mathbb{N}[q,t]$ for all $r\geq 1$. We establish a recursion showing that $(-1)^{|\mu|-\ell(\mu)}m_\mu$ has an expansion with coefficients in $\mathbb{Q}_{\geq 0}[q]$ in the symmetric functions $C_\alpha(1)$, where $C_a$ denotes the operator introduced by Haglund, Morse and Zabrocki. Combining this expansion with the compositional shuffle theorems of Carlsson--Mellit and Mellit, and with the Schur positivity of LLT polynomials, completes the proof. The same method, using the $e$-positivity of column LLT polynomials after the substitution $q\mapsto q+1$, also gives an $e$-positive analogue.