{"paper":{"title":"Partition division maps, symmetric functions and positivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A linear map dividing a partition by k sends Schur functions to Schur-positive symmetric functions.","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Lilan Dai, Per Alexandersson","submitted_at":"2026-04-28T09:49:11Z","abstract_excerpt":"We study a linear map on symmetric functions that ``divides'' a partition by a positive integer $k$, sending a Schur function indexed by a partition of $kn$ to a symmetric function indexed by partitions of $n$. We determine its Schur expansion explicitly for Schur and skew Schur functions, showing that the coefficients are enumerated by a new family of combinatorial objects, called $k$-Yamanouchi tableaux, which generalize the classical ballot (Yamanouchi) tableaux appearing in the Littlewood--Richardson rule.\n  We also study the images of elementary symmetric functions under this map, derive "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that the image of this map is always Schur-positive, meaning it expands in the Schur basis with nonnegative integer coefficients. These coefficients are enumerated by a new family of combinatorial objects, called k-Yamanouchi tableaux.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The linear map is well-defined on the Schur basis and the combinatorial interpretation via k-Yamanouchi tableaux correctly enumerates the coefficients without hidden cancellations or sign issues.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A new partition-division map on symmetric functions produces Schur-positive outputs enumerated by k-Yamanouchi tableaux.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A linear map dividing a partition by k sends Schur functions to Schur-positive symmetric functions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c08c3f9e3a8e3619bbdbef55abcf4c12d9e873087c436999efadede7aa409e3b"},"source":{"id":"2604.25440","kind":"arxiv","version":2},"verdict":{"id":"7fe08852-876b-4d97-8b12-15be69c580f2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T15:45:55.568172Z","strongest_claim":"We prove that the image of this map is always Schur-positive, meaning it expands in the Schur basis with nonnegative integer coefficients. These coefficients are enumerated by a new family of combinatorial objects, called k-Yamanouchi tableaux.","one_line_summary":"A new partition-division map on symmetric functions produces Schur-positive outputs enumerated by k-Yamanouchi tableaux.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The linear map is well-defined on the Schur basis and the combinatorial interpretation via k-Yamanouchi tableaux correctly enumerates the coefficients without hidden cancellations or sign issues.","pith_extraction_headline":"A linear map dividing a partition by k sends Schur functions to Schur-positive symmetric functions."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.25440/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T04:40:08.417300Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T21:08:34.337187Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"86eabeaaf5ce90d288721a5b15f78552ec77e107e51a3f0cc57f9364f82c4591"},"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"}