{"paper":{"title":"Refinement of Hikita's $e$-positivity theorem via Abreu--Nigro's $g$-functions and restricted modular law","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Byung-Hak Hwang, Donghyun Kim, Jaeseong Oh, Jang Soo Kim, JiSun Huh","submitted_at":"2025-04-12T08:17:45Z","abstract_excerpt":"We study the symmetric functions \\( g_{\\mm,k}(x;q) \\), introduced by\n  Abreu and Nigro for a Hessenberg function \\( \\mm \\) and a positive\n  integer \\( k \\), which refine the chromatic symmetric function.\n  Building on Hikita's recent breakthrough on the Stanley--Stembridge\n  conjecture, we prove the \\( e \\)-positivity of \\( g_{\\mm,k}(x;1) \\),\n  refining Hikita's result. We also provide a Schur expansion of the\n  sum \\( \\sum_{k=1}^n e_k(x) g_{\\mm,n-k}(x;q) \\) in terms of\n  \\( P \\)-tableaux with 1 in the upper-left corner. We introduce a\n  restricted version of the modular law as our main tool. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.09123","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.09123/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}