Exploring a Delta Schur Conjecture

In \cite{HRW15}, Haglund, Remmel, Wilson state a conjecture which predicts a purely combinatorial way of obtaining the symmetric function $\Delta_{e_k}e_n$. It is called the Delta Conjecture. It was recently proved in \cite{GHRY} that the Delta Conjecture is true when either $q=0$ or $t=0$. In this paper we complete a work initiated by Remmel whose initial aim was to explore the symmetric function $\Delta_{s_\nu} e_n$ by the same methods developed in \cite{GHRY}. Our first need here is a method for constructing a symmetric function that may be viewed as a "combinatorial side" for the symmetric function $\Delta_{s_\nu} e_n$ for $t=0$. Based on what was discovered in \cite{GHRY} we conjectured such a construction mechanism. We prove here that in the case that $\nu=(m-k,1^k)$ with $1\le m< n$ the equality of the two sides can be established by the same methods used in \cite{GHRY}. While this work was in progress, we learned that Rhodes and Shimozono had previously constructed also such a "combinatorial side". Very recently, Jim Haglund was able to prove that their conjecture follows from the results in \cite{GHRY}. We show here that an appropriate modification of the Haglund arguments proves that the polynomial $\Delta_{s_\nu}e_n$ as well as the Rhoades-Shimozono "combinatorial side" have a plethystic evaluation with hook Schur function expansion.