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arXiv · 2610.34772

The $e$-expansion of integral form Macdonald polynomials

Jim Haglund·Jang Soo Kim·Jaeseong Oh·Meesue Yoo
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Abstract

In this paper, we investigate the expansion of the integral form Macdonald polynomials $J_μ(x;q,t)$ in the elementary symmetric function basis. Motivated by the connection between chromatic symmetric functions and LLT polynomials, we conjecture the $e$-positivity of a normalization of $J_μ(x;q,t)$. We verify the conjecture when $μ$ is either a hook or a two-column partition. In the hook case, we express the $e$-coefficients as weighted sums over compositions. In the two-column case, we derive explicit formulas for the $e$-coefficients.

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Jim Haglund, Jang Soo Kim, Jaeseong Oh, Meesue Yoo. 2026-10-26. The $e$-expansion of integral form Macdonald polynomials. https://arxiv.org/abs/2610.34772

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