Delta and Theta Operator Expansions

TL;DR

This paper provides elementary symmetric function expansions for certain Delta and Theta operators at t=1, introducing gamma-parking functions and lattice gamma-parking functions, with implications for the elementary basis expansion of symmetric functions.

Contribution

It introduces new expansions for Delta and Theta operators at t=1 using gamma-parking functions, advancing understanding of symmetric function bases and proposing an e-positivity conjecture.

Findings

Elementary basis expansion at t=1 for specific symmetric functions

Introduction of gamma-parking functions and lattice gamma-parking functions

Proposed e-positivity conjecture for general t

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.