Schur Function Expansions and the Rational Shuffle Theorem

Abstract

Gorsky and Negut [9] 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 [15]. Later, Bergeron, Garsia, Leven and Xin [3] 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 [12]. The Shuffle Conjecture was proved in 2015 by Carlsson and Mellit [4] and full Rational Shuffle Conjecture was later proved by Mellit [19]. 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 [17]. 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.