{"paper":{"title":"The Schur positivity of $\\nabla m_\\mu$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dun Qiu, Minhao Zhang","submitted_at":"2026-07-01T13:41:04Z","abstract_excerpt":"Bergeron, Garsia, Haiman and Tesler conjectured in 1999 that, for all partitions $\\mu,\\lambda\\vdash n$, the polynomial $(-1)^{|\\mu|-\\ell(\\mu)}\\langle \\nabla m_\\mu, s_\\lambda\\rangle$ has nonnegative integer coefficients, where $\\nabla$ is the Bergeron--Garsia nabla operator, which acts diagonally on the modified Macdonald basis, and $m_\\mu$ is the monomial symmetric function. In this article, we prove this conjecture, and more generally that $(-1)^{|\\mu|-\\ell(\\mu)}\\langle\\nabla^r m_\\mu,s_\\lambda\\rangle\\in\\mathbb{N}[q,t]$ for all $r\\geq 1$. We establish a recursion showing that $(-1)^{|\\mu|-\\ell"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00940","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/2607.00940/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"}