Delta and Theta Operator Expansions (2203.10342v4)

Abstract: We give an elementary symmetric function expansion for $M\Delta_{m_\gamma e_1}\Pi e_\lambda^{\ast}$ and $M\Delta_{m_\gamma e_1}\Pi s_\lambda^{\ast}$ when $t=1$ in terms of what we call $\gamma$-parking functions and lattice $\gamma$-parking functions. Here, $\Delta_F$ and $\Pi$ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$. Our main results in turn give an elementary basis expansion at $t=1$ for symmetric functions of the form $M \Delta_{Fe_1} \Theta_{G} J$ whenever $F$ is expanded in terms of monomials, $G$ is expanded in terms of the elementary basis, and $J$ is expanded in terms of the modified elementary basis ${\Pi e_\lambda^\ast}_\lambda$. Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving an $e$-positivity conjecture for when $t$ is not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function.