The Schur positivity of $\mathrm{\nabla }{m}_{\mu }$

Bergeron, Garsia, Haiman and Tesler conjectured in 1999 that, for all partitions $\mu ,\lambda ⊢n$, the polynomial $(-1{)}^{\mathrm{\mid }\mu \mathrm{\mid }-\mathrm{\ell }(\mu )}⟨\mathrm{\nabla }{m}_{\mu },s_{\lambda }⟩$ has nonnegative integer coefficients, where $\mathrm{\nabla }$ 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 $(-1{)}^{\mathrm{\mid }\mu \mathrm{\mid }-\mathrm{\ell }(\mu )}⟨{\mathrm{\nabla }}^r{m}_{\mu },s_{\lambda }⟩\in \mathbb{N}[q,t]$ for all $r\ge 1$. We establish a recursion showing that $(-1{)}^{\mathrm{\mid }\mu \mathrm{\mid }-\mathrm{\ell }(\mu )}{m}_{\mu }$ has an expansion with coefficients in ${\mathbb{Q}}_{\ge 0}[q]$ 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.