Schur Function Expansions and the Rational Shuffle Conjecture
arXiv: Combinatorics(2018)
摘要
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 expansion of $Q_{m,n}(-1)^n$ in terms of the fundamental quasi-symmetric functions are given by polynomials introduced by Hikita. Later, Bergeron, Garsia, Leven and Xin extended and refined the conjectures of Gorsky and Negut to give a combinatorial interpretation of the coefficients that arise in the expansion of $Q_{m,n}(-1)^n$ in terms of the fundamental quasi-symmetric functions for arbitrary $m$ and $n$ which we will call the Rational Shuffle Conjecture. In the special case $Q_{n+1,n}(-1)^{n}$, the Rational Shuffle Conjecture becomes the Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov. The Shuffle Conjecture was proved in 2015 by Carlsson and Mellit and full Rational Shuffle Conjecture was later proved by Mellit. The main goal of this paper is to study the combinatorics of the coefficients that arise in the Schur function expansion of $Q_{m,n}(-1)^n$ in certain special cases. Leven gave a combinatorial proof of the Schur function expansion of $Q_{2,2n+1}(-1)^{2n+1}$ and $Q_{2n+1,2}(-1)^2$. In this paper, we explore several symmetries in the combinatorics of the coefficients that arise in the Schur function expansion of $Q_{m,n}(-1)^n$. Especially, we study the hook-shaped Schur function coefficients, and the Schur function expansion of $Q_{m,n}(-1)^n$ in the case where $m$ or $n$ equals $3$.
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