2026/07/01 by Dun Qiu, Minhao Zhang
#math.CO
Bergeron, Garsia, Haiman and Tesler conjectured in 1999 that, for all partitions μ,λ\vdash n, the polynomial (-1)|μ|-ℓ(μ)⟨ ∇ mμ, sλ⟩ has nonnegative integer coefficients, where ∇ is the Bergeron--Garsia nabla operator, which acts diagonally on the modified Macdonald basis, and mμ is the monomial symmetric function. In this article, we prove this conjecture, and more generally that (-1)|μ|-ℓ(μ)⟨∇r mμ,sλ⟩∈ℕ[q,t] for all r≥ 1. We establish a recursion showing that (-1)|μ|-ℓ(μ)mμ has an expansion with coefficients in ℚ≥ 0[q] in the symmetric functions Cα(1), where Ca denotes the operator introduced by Haglund, Morse and Zabrocki. Combining this expansion with the compositional shuffle theorems of Carlsson--Mellit and Mellit, and with the Schur positivity of LLT polynomials, completes the proof. The same method, using the e-positivity of column LLT polynomials after the substitution q↦ q+1, also gives an e-positive analogue.