A solution to Butler's positivity conjecture

Abstract

Let $λ, μ, ν$ be distinct partitions such that $λ, μ\subset ν$ and $|ν/λ|=|ν/μ|=1$. We prove Butler's positivity conjecture posed in 1994: the expansion of the Macdonald intersection polynomial \[ \frac{T_λ\widetilde{H}_μ(X;q,t)-T_μ\widetilde{H}_λ(X;q,t)}{T_λ-T_μ} \] in terms of the Schur function basis has coefficients in $\mathbb{Z}_{\geq 0}[q,t]$.

Disclosure

“eme of points in the affine plane. Acknowledgements. The first author was supported by the National Natural Science Foundation of China (No. 12371329) and the Fundamental Research Funds for the Central Universities (No. 63263094). We used ChatGPT 5.6 during the preparation of this work.”

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Classification

Drafting limited passages

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5

Verified

2026-08-23T05:10:55.004641Z

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