The Schur positivity of $\nabla m_μ$

Abstract

Bergeron, Garsia, Haiman and Tesler conjectured in 1999 that, for all partitions $μ,λ\vdash n$, the polynomial $(-1)^{|μ|-\ell(μ)}\langle \nabla m_μ, s_λ\rangle$ has nonnegative integer coefficients, where $\nabla$ is the Bergeron--Garsia nabla operator, which acts diagonally on the modified Macdonald basis, and $m_μ$ is the monomial symmetric function. In this article, we prove this conjecture, and more generally that $(-1)^{|μ|-\ell(μ)}\langle\nabla^r m_μ,s_λ\rangle\in\mathbb{N}[q,t]$ for all $r\geq 1$. We establish a recursion showing that $(-1)^{|μ|-\ell(μ)}m_μ$ has an expansion with coefficients in $\mathbb{Q}_{\geq 0}[q]$ in the symmetric functions $C_α(1)$, 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.