Toward Butler's conjecture

Abstract

For a partition $\hat{I}½$, let $\hat{I}»,\hat{I}¼\subseteq \hat{I}½$ be two distinct partitions such that $|\hat{I}½/\hat{I}»|=|\hat{I}½/\hat{I}¼|=1$. Butler conjectured that the divided difference $I_{\hat{I}»,\hat{I}¼}[X;q,t]=(T_{\hat{I}}»{\tilde{H}}_{\hat{I}}¼[X;q,t]-T_{\hat{I}}¼{\tilde{H}}_{\hat{I}}»[X;q,t])/(T_{\hat{I}}»-T_{\hat{I}}¼)$ of modified Macdonald polynomials of two partitions $\hat{I}»$ and $\hat{I}¼$ is Schur positive. By introducing a new LLT equivalence called column exchange rule, we give a combinatorial formula for $I_{\hat{I}»,\hat{I}¼}[X;q,t]$, which is a positive monomial expansion. We also prove Butler's conjecture for some special cases.

Proceedings of the London Mathematical Society to appear