Toward Butler's conjecture

The celebrated Haglund–Haiman–Loehr (HHL) formula provides an explicit monomial expansion of the Macdonald polynomials. In 1994, Butler introduced a refinement of the Macdonald polynomial and conjectured its Schur positivity. According to the Science Fiction conjecture by Bergeron and Garsia, this refinement represents the “intersection” of Macdonald polynomials. In this work, we introduce a novel combinatorial tool, the column exchange rule , which enables us to derive a positive monomial expansion for Butler's symmetric function , thereby refining the HHL formula. Additionally, we prove Butler's conjecture on the Schur positivity of in specific cases.