Refinement of Hikita's e-positivity theorem via Abreu--Nigro's g-functions and restricted modular law

2025/04/12 by JiSun Huh, Byung-Hak Hwang, Huh, JiSun +6 · 1 citation
Mathematics · #Advanced Mathematical Theories #Approximation Theory and Sequence Spaces #Combinatorics (math.CO) #FOS: Mathematics #Fuzzy Systems and Optimization

paper · pdf · doi:10.48550/arxiv.2504.09123

openalex publication_date 2025/04/12 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

We study the symmetric functions \( g_\mm,k(x;q) \), introduced by Abreu and Nigro for a Hessenberg function \( \mm \) and a positive integer \( k \), which refine the chromatic symmetric function. Building on Hikita's recent breakthrough on the Stanley--Stembridge conjecture, we prove the \( e \)-positivity of \( g_\mm,k(x;1) \), refining Hikita's result. We also provide a Schur expansion of the sum \( ∑_k=1ⁿ e_k(x) g_\mm,n-k(x;q) \) in terms of \( P \)-tableaux with 1 in the upper-left corner. We introduce a restricted version of the modular law as our main tool. Then, we show that any function satisfying the restricted modular law is determined by its values on disjoint unions of path graphs.

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• Toward Lower Bounds for Chromatic Symmetric Functions in the Elementary Basis

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