A solution to Butler's positivity conjecture

Abstract

Let $λ, μ, ν$ be distinct partitions such that $λ, μ\subset ν$ and $|ν/λ|=|ν/μ|=1$. We prove Butler's positivity conjecture posed in 1994: the expansion of the Macdonald intersection polynomial \[ \frac{T_λ\widetilde{H}_μ(X;q,t)-T_μ\widetilde{H}_λ(X;q,t)}{T_λ-T_μ} \] in terms of the Schur function basis has coefficients in $\mathbb{Z}_{\geq 0}[q,t]$.