On the Schur positivity of ${\Delta }_{e_2}{e}_n[X]$

Abstract

Let $\mathbb{N}$ denote the set of non-negative integers. Haglund, Wilson, and the second author have conjectured that the coefficient of any Schur function $s_{\lambda }[X]$ in ${\Delta }_{e_k}{e}_n[X]$ is a polynomial in $\mathbb{N}[q,t]$. We present four proofs of a stronger statement in the case $k=2$; We show that the coefficient of any Schur function $s_{\lambda }[X]$ in ${\Delta }_{e_2}{e}_n[X]$ has a positive expansion in terms of $q,t$-analogs.