Schur Function Expansions and the Rational Shuffle Conjecture

Dun Qiu ; Jeffrey B. Remmel

Séminaire lotharingien de combinatoire, 78B (2017)

Résumé

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 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. The rational shuffle conjecture was later proved by Mellit in 2016. 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 the case where m or n equals 3.