{"paper":{"title":"$k$-Schur expansions of Catalan functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.QA"],"primary_cat":"math.CO","authors_text":"Anna Pun, Daniel Summers, Jennifer Morse, Jonah Blasiak","submitted_at":"2018-11-06T17:02:56Z","abstract_excerpt":"We make a broad conjecture about the $k$-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which address the $k$-Schur expansion of (1) Hall-Littlewood polynomials, proving the $q=0$ case of the strengthened Macdonald positivity conjecture of Lapointe, Lascoux, and Morse; (2) the product of a Schur function and a $k$-Schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum coh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.02490","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":""},"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"}