Work

Toward Butler's conjecture

Year: 2026

Type: article

Abstract: 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.

Source: Proceedings of the London Mathematical Society

DOI: 10.1112/plms.70127

Authors: Donghyun Kim, Seung Jin Lee, Jaeseong Oh

Institutions: Seoul National University, Sungkyunkwan University

Language: English

Topic: Advanced Combinatorial Mathematics

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funder: National Research Foundation of Korea

Awards: G1068867740, G586508345, G8644082832