Schur function expansions and the Rational Shuffle Theorem

Gorsky and Negut introduced operators $Q_{m,n}$ on symmetric functions and conjectured that, in the case where $m$ and $n$ are relatively prime, the expression ${Q}_{m,n}(1)$ is given by the Hikita polynomial ${H}_{m,n}[X;q,t]$. Later, Bergeron-Garsia-Leven-Xin extended and refined the conjectures of ${Q}_{m,n}(1)$ for arbitrary $m$ and $n$ which we call the Extended Rational Shuffle Conjecture. In the special case ${Q}_{n+1,n}(1)$, the Rational Shuffle Conjecture becomes the Shuffle Conjecture of Haglund-Haiman-Loehr-Remmel-Ulyanov, which was proved in 2015 by Carlsson and Mellit as the Shuffle Theorem. The Extended Rational Shuffle Conjecture was later proved by Mellit as the Extended Rational Shuffle Theorem. The main goal of this paper is to study the combinatorics of the coefficients that arise in the Schur function expansion of ${Q}_{m,n}(1)$ in certain special cases. Leven gave a combinatorial proof of the Schur function expansion of ${Q}_{2,2n+1}(1)$ and ${Q}_{2n+1,2}(1)$. In this paper, we explore several symmetries in the combinatorics of the coefficients that arise in the Schur function expansion of ${Q}_{m,n}(1)$. Especially, we study the hook-shaped Schur function coefficients, and the Schur function expansion of ${Q}_{m,n}(1)$ in the case where $m$ or $n$ equals $3$.