{"id":"W2032758875","doi":"10.1142/s0219498809003485","title":"THE HOPF ALGEBRAS OF SYMMETRIC FUNCTIONS AND QUASI-SYMMETRIC FUNCTIONS IN NON-COMMUTATIVE VARIABLES ARE FREE AND CO-FREE","year":2009,"lang":"en","type":"article","venue":"Journal of Algebra and Its Applications","topic":"Advanced Combinatorial Mathematics","field":"Mathematics","cited_by":41,"is_retracted":false,"has_abstract":true,"ca_institutions":"York University","funders":"","keywords":"Hopf algebra; Representation theory of Hopf algebras; Coproduct; Symmetric function; Basis (linear algebra); Algebra over a field; Monomial; Product (mathematics); Ring of symmetric functions","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":[0.001211328,0.0007703203,0.0007754014,0.001604946,0.001547965,0.002759762,0.0008345032,0.0005317666,0.00640947],"category_scores_gemma":[0.00153196,0.0003393833,0.0007279925,0.001068944,0.004180483,0.005202688,0.002270428,0.001649905,0.001225223],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0006866218,"about_ca_system_score_gemma":0.0009743673,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000721593,"about_ca_topic_score_gemma":0.0008426796,"domain_scores_codex":[0.9990625,0.0001542803,0.00006031357,0.0001672132,0.0003777407,0.0001779579],"domain_scores_gemma":[0.9987175,0.000315172,0.0002125809,0.0001697962,0.0003394753,0.0002454265],"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":[0.00002044349,0.00001459691,0.0001780942,0.00002563956,0.000008440311,0.0000878116,0.0002159598,0.000345602,0.001583694,0.9923162,0.0001781006,0.005025362],"study_design_scores_gemma":[0.000009938209,0.00001793392,0.0002559132,0.00001075216,0.00001233553,0.0001141361,0.0001498636,0.002055313,0.003841642,0.9894469,0.004070479,0.00001476142],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.5090438,0.002576529,0.3900416,0.001371777,0.0005977278,0.0001226974,0.0003815836,0.0002926695,0.09557153],"genre_scores_gemma":[0.9439974,0.0008409544,0.02906506,0.0001852626,0.0002215777,0.00007692655,0.0002177334,0.00007466279,0.02532046],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.00640947,"threshold_uncertainty_score":0.02144182,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.01945149051198161,"score_gpt":0.28804752458313,"score_spread":0.2685960340711484,"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."}}