{"id":"W1828251690","doi":"10.37236/4139","title":"Schubert Polynomials and $k$-Schur Functions","year":2014,"lang":"en","type":"preprint","venue":"The Electronic Journal of Combinatorics","topic":"Advanced Combinatorial Mathematics","field":"Mathematics","cited_by":1,"is_retracted":false,"has_abstract":true,"ca_institutions":"York University","funders":"","keywords":"Schur polynomial; Schur algebra; Mathematics; Schubert calculus; Grassmannian; Schur's theorem; Schubert polynomial; Schur's lemma; Combinatorics; Partially ordered set; Schur product theorem; Pure mathematics; Schur decomposition; Schur complement; Macdonald polynomials; Orthogonal polynomials; Classical orthogonal polynomials; Gegenbauer polynomials","routes":{"ca_aff":true,"ca_fund":false,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0006177477,0.0006655447,0.0004565256,0.001456393,0.001453758,0.002422762,0.0005373819,0.000650877,0.005613097],"category_scores_gemma":[0.001907579,0.0002367768,0.0005614475,0.001059332,0.00379949,0.003914474,0.001719084,0.00180586,0.0006984653],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0007724366,"about_ca_system_score_gemma":0.000485522,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0006001936,"about_ca_topic_score_gemma":0.0005060842,"domain_scores_codex":[0.9993203,0.0001888844,0.00002570501,0.0001288459,0.000177269,0.0001590648],"domain_scores_gemma":[0.9991642,0.0002855246,0.0001852396,0.00006220011,0.0001482438,0.0001545659],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.00001495016,0.00000710851,0.00012324,0.0000146134,0.000001585261,0.00005958926,0.0001399888,0.000364447,0.0006345246,0.9957801,0.0002373524,0.002622381],"study_design_scores_gemma":[0.000006839914,0.00002750637,0.0002759022,0.00001259975,0.000004667651,0.0001950244,0.0002429718,0.004024496,0.001085915,0.9894438,0.004662511,0.00001780843],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.6921692,0.00123156,0.1864922,0.001956665,0.0003839745,0.00007442707,0.0002068249,0.0001640488,0.1173211],"genre_scores_gemma":[0.9717941,0.000691059,0.01560027,0.000210379,0.000449979,0.00005441359,0.0001298892,0.00004715773,0.01102289],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.005613097,"threshold_uncertainty_score":0.01877767,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.02277605596070487,"score_gpt":0.2985247372467303,"score_spread":0.2757486812860255,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}