Toward Butler's conjecture

For a partition $\nu$, let $\lambda ,\mu \subseteq \nu$ be two distinct partitions such that $\mathrm{\mid }\nu \mathrm{/}\lambda \mathrm{\mid }=\mathrm{\mid }\nu \mathrm{/}\mu \mathrm{\mid }=1$. Butler conjectured that the divided difference ${\mathrm{I}}_{\lambda ,\mu }[X;q,t]=(T_{\lambda }{\tilde{H}}_{\mu }[X;q,t]-T_{\mu }{\tilde{H}}_{\lambda }[X;q,t])\mathrm{/}(T_{\lambda }-T_{\mu })$ of modified Macdonald polynomials of two partitions $\lambda$ and $\mu$ is Schur positive. By introducing a new LLT equivalence called column exchange rule, we give a combinatorial formula for ${\mathrm{I}}_{\lambda ,\mu }[X;q,t]$, which is a positive monomial expansion. We also prove Butler’s conjecture for some special cases.