Year: 2026
Type: preprint
Abstract: Bergeron, Garsia, Haiman, and Tesler (1999) introduced two Schur positivity conjectures concerning the action of the $\nabla$ operator on modified Hall--Littlewood polynomials. Although the proofs of the celebrated compositional shuffle conjecture and its rational generalization, along with various symmetric function identities, have largely resolved these conjectures in the affirmative, we provide an elegant alternative proof driven by $q$-series techniques. Through this approach, we derive explicit Schur function expansions for two-column partitions and extend our results to arbitrary positive powers $\nabla^k$.
Source: arXiv (Cornell University)
Author: Qu, Menghao
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162029971