Séminaire Lotharingien de Combinatoire, 89B.83 (2023), 12 pp.

Donghyun Kim, Seung Jin Lee and Jaeseong Oh

Towards Butler's Conjecture

Abstract. For a partition ν, let λ,μ ⊆ ν be two distinct partitions such that |ν/λ| = |ν/μ| = 1. Butler conjectured that the divided difference Iλ,μ[X;q,t] = (TH~μ[X;q,t] - TμH~ λ[X;q,t])/(Tλ-Tμ) of modified Macdonald polynomials of two partitions λ and μ is Schur positive. By introducing a new LLT equivalence called column exchange rule, we give a combinatorial formula for Iλ,μ}[X;q,t], which is a positive monomial expansion. We also prove Butler's conjecture for some special cases.

Received: November 15, 2022. Accepted: February 20, 2023. Final version: April 1, 2023.

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