A solution to Butler's positivity conjecture

Abstract

Let $\lambda ,\mu ,\nu$ be distinct partitions such that $\lambda ,\mu \subset \nu$ and $|\nu /\lambda |=|\nu /\mu |=1$. 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 ${\mathbb{Z}}_{\ge 0}[q,t]$.

6 pages