{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:JOLJOAY4ARSGBVL5OHD65FTNIW","short_pith_number":"pith:JOLJOAY4","schema_version":"1.0","canonical_sha256":"4b9697031c046460d57d71c7ee966d45bfdb6269cff9da9fd1e4c23bc0e73926","source":{"kind":"arxiv","id":"2607.00940","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.00940","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-01T13:41:04Z","cross_cats_sorted":[],"title_canon_sha256":"11a9ef84952d6aeb7776c0a7f12f5812faa763a2462a2b6cf2c3d7f6e486cc12","abstract_canon_sha256":"1ccd91b85932bfcc99a7e880fba09a320446c75dd446f47e332cec07ce556db8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-02T01:18:24.069865Z","signature_b64":"nQGdWoY82I7zHFQxpzBxbdo3YrFVC3lCX3xL/9iVRAJFrbjdMkw8SK5pC9C5Iwi6tyMt7SVoK4nYU8xk8LjlAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b9697031c046460d57d71c7ee966d45bfdb6269cff9da9fd1e4c23bc0e73926","last_reissued_at":"2026-07-02T01:18:24.069524Z","signature_status":"signed_v1","first_computed_at":"2026-07-02T01:18:24.069524Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.00940","created_at":"2026-07-02T01:18:24.069577+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.00940v1","created_at":"2026-07-02T01:18:24.069577+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.00940","created_at":"2026-07-02T01:18:24.069577+00:00"},{"alias_kind":"pith_short_12","alias_value":"JOLJOAY4ARSG","created_at":"2026-07-02T01:18:24.069577+00:00"},{"alias_kind":"pith_short_16","alias_value":"JOLJOAY4ARSGBVL5","created_at":"2026-07-02T01:18:24.069577+00:00"},{"alias_kind":"pith_short_8","alias_value":"JOLJOAY4","created_at":"2026-07-02T01:18:24.069577+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2607.06351","citing_title":"Schur positivity of nabla on Petrie symmetric functions","ref_index":22,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JOLJOAY4ARSGBVL5OHD65FTNIW","json":"https://pith.science/pith/JOLJOAY4ARSGBVL5OHD65FTNIW.json","graph_json":"https://pith.science/api/pith-number/JOLJOAY4ARSGBVL5OHD65FTNIW/graph.json","events_json":"https://pith.science/api/pith-number/JOLJOAY4ARSGBVL5OHD65FTNIW/events.json","paper":"https://pith.science/paper/JOLJOAY4"},"agent_actions":{"view_html":"https://pith.science/pith/JOLJOAY4ARSGBVL5OHD65FTNIW","download_json":"https://pith.science/pith/JOLJOAY4ARSGBVL5OHD65FTNIW.json","view_paper":"https://pith.science/paper/JOLJOAY4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.00940&json=true","fetch_graph":"https://pith.science/api/pith-number/JOLJOAY4ARSGBVL5OHD65FTNIW/graph.json","fetch_events":"https://pith.science/api/pith-number/JOLJOAY4ARSGBVL5OHD65FTNIW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JOLJOAY4ARSGBVL5OHD65FTNIW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JOLJOAY4ARSGBVL5OHD65FTNIW/action/storage_attestation","attest_author":"https://pith.science/pith/JOLJOAY4ARSGBVL5OHD65FTNIW/action/author_attestation","sign_citation":"https://pith.science/pith/JOLJOAY4ARSGBVL5OHD65FTNIW/action/citation_signature","submit_replication":"https://pith.science/pith/JOLJOAY4ARSGBVL5OHD65FTNIW/action/replication_record"}},"created_at":"2026-07-02T01:18:24.069577+00:00","updated_at":"2026-07-02T01:18:24.069577+00:00"}