{"id":"https://openalex.org/W1483701399","doi":"https://doi.org/10.37236/1814","title":"A Positive Proof of the Littlewood-Richardson Rule using the Octahedron Recurrence","display_name":"A Positive Proof of the Littlewood-Richardson Rule using the Octahedron Recurrence","publication_year":2004,"publication_date":"2004-09-13","ids":{"openalex":"https://openalex.org/W1483701399","doi":"https://doi.org/10.37236/1814","mag":"1483701399"},"language":"en","primary_location":{"id":"doi:10.37236/1814","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1814","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1r61/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1r61/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5074655749","display_name":"Allen Knutson","orcid":null},"institutions":[{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Allen Knutson","raw_affiliation_strings":["Mathematics Department, UC Berkeley, Berkeley, California"],"affiliations":[{"raw_affiliation_string":"Mathematics Department, UC Berkeley, Berkeley, California","institution_ids":["https://openalex.org/I95457486"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5066728260","display_name":"Terence Tao","orcid":"https://orcid.org/0000-0002-0140-7641"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Terence Tao","raw_affiliation_strings":["Mathematics Department, UCLA, Los Angeles, California"],"affiliations":[{"raw_affiliation_string":"Mathematics Department, UCLA, Los Angeles, California","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5108377818","display_name":"Christopher T. Woodward","orcid":null},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Christopher Woodward","raw_affiliation_strings":["Mathematics Department, Rutgers University, New Brunswick, New Jersey"],"affiliations":[{"raw_affiliation_string":"Mathematics Department, Rutgers University, New Brunswick, New Jersey","institution_ids":["https://openalex.org/I102322142"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5074655749"],"corresponding_institution_ids":["https://openalex.org/I95457486"],"apc_list":null,"apc_paid":null,"fwci":7.7769,"has_fulltext":true,"cited_by_count":70,"citation_normalized_percentile":{"value":0.9796413,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"11","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11680","display_name":"Advanced Algebra and Geometry","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/bijection","display_name":"Bijection","score":0.7814467549324036},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7343705892562866},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7296454310417175},{"id":"https://openalex.org/keywords/ring","display_name":"Ring (chemistry)","score":0.7006614208221436},{"id":"https://openalex.org/keywords/octahedron","display_name":"Octahedron","score":0.5451241731643677},{"id":"https://openalex.org/keywords/tensor","display_name":"Tensor (intrinsic definition)","score":0.45509833097457886},{"id":"https://openalex.org/keywords/ideal","display_name":"Ideal (ethics)","score":0.4250301718711853},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.21525084972381592},{"id":"https://openalex.org/keywords/crystallography","display_name":"Crystallography","score":0.13411137461662292},{"id":"https://openalex.org/keywords/crystal-structure","display_name":"Crystal structure","score":0.07597336173057556},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.06265825033187866}],"concepts":[{"id":"https://openalex.org/C24424167","wikidata":"https://www.wikidata.org/wiki/Q180907","display_name":"Bijection","level":2,"score":0.7814467549324036},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7343705892562866},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7296454310417175},{"id":"https://openalex.org/C2780378348","wikidata":"https://www.wikidata.org/wiki/Q25351438","display_name":"Ring (chemistry)","level":2,"score":0.7006614208221436},{"id":"https://openalex.org/C198819137","wikidata":"https://www.wikidata.org/wiki/Q188884","display_name":"Octahedron","level":3,"score":0.5451241731643677},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.45509833097457886},{"id":"https://openalex.org/C2776639384","wikidata":"https://www.wikidata.org/wiki/Q840396","display_name":"Ideal (ethics)","level":2,"score":0.4250301718711853},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.21525084972381592},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.13411137461662292},{"id":"https://openalex.org/C115624301","wikidata":"https://www.wikidata.org/wiki/Q895901","display_name":"Crystal structure","level":2,"score":0.07597336173057556},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.06265825033187866},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/1814","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1814","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1r61/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.37236/1814","is_oa":true,"landing_page_url":"https://doi.org/10.37236/1814","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1r61/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1441707171","display_name":null,"funder_award_id":"Fellow","funder_id":"https://openalex.org/F4320306079","funder_display_name":"David and Lucile Packard Foundation"},{"id":"https://openalex.org/G1948944922","display_name":"Generalized cohomology theories of flag manifolds, and other manifolds","funder_award_id":"0072667","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4577819096","display_name":"Moduli spaces of flat connections and Hamiltonian actions of loop groups","funder_award_id":"9971357","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4641068160","display_name":null,"funder_award_id":"Packard","funder_id":"https://openalex.org/F4320306079","funder_display_name":"David and Lucile Packard Foundation"},{"id":"https://openalex.org/G6894402473","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8832243575","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320306079","funder_display_name":"David and Lucile Packard Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306079","display_name":"David and Lucile Packard Foundation","ror":"https://ror.org/032atxq54"},{"id":"https://openalex.org/F4320308569","display_name":"Clay Mathematics Institute","ror":"https://ror.org/02chwk660"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1483701399.pdf","grobid_xml":"https://content.openalex.org/works/W1483701399.grobid-xml"},"referenced_works_count":22,"referenced_works":["https://openalex.org/W1503979395","https://openalex.org/W1657156448","https://openalex.org/W1664429083","https://openalex.org/W1807235826","https://openalex.org/W2038782505","https://openalex.org/W2060244141","https://openalex.org/W2087997360","https://openalex.org/W2100445436","https://openalex.org/W2132372851","https://openalex.org/W2135893164","https://openalex.org/W2147636812","https://openalex.org/W2152891537","https://openalex.org/W2952500876","https://openalex.org/W2963000857","https://openalex.org/W2965314093","https://openalex.org/W4240060697","https://openalex.org/W6602880896","https://openalex.org/W6606807746","https://openalex.org/W6630312485","https://openalex.org/W6637972615","https://openalex.org/W6638883921","https://openalex.org/W6682042534"],"related_works":["https://openalex.org/W4287020903","https://openalex.org/W2952259524","https://openalex.org/W2279732055","https://openalex.org/W2995857455","https://openalex.org/W4288805235","https://openalex.org/W2998840281","https://openalex.org/W2027277382","https://openalex.org/W1508013399","https://openalex.org/W3181135859","https://openalex.org/W4302399754"],"abstract_inverted_index":{"We":[0,43],"define":[1],"the":[2,45,62,70,79,86,89,95,101,123],"hive":[3,71],"ring,":[4],"which":[5],"has":[6],"a":[7,66,76,119],"basis":[8],"indexed":[9],"by":[10,20],"dominant":[11],"weights":[12],"for":[13,122],"$GL_n({\\Bbb":[14,83],"C})$,":[15],"and":[16,61,103],"structure":[17],"constants":[18],"given":[19],"counting":[21],"hives":[22],"[Knutson-Tao,":[23],"\"The":[24],"honeycomb":[25,87],"model":[26],"of":[27,82,94,100],"$GL_n$":[28],"tensor":[29],"products\"]":[30],"(or":[31],"equivalently":[32],"honeycombs,":[33],"or":[34],"BZ":[35],"patterns":[36],"[Berenstein-Zelevinsky,":[37],"\"Involutions":[38],"on":[39],"Gel$'$fand-Tsetlin":[40],"schemes$\\dots$":[41],"\"]).":[42],"use":[44,116],"octahedron":[46,90],"rule":[47,91],"from":[48,106],"[Robbins-Rumsey,":[49],"\"Determinants$\\dots$\"]":[50],"to":[51,78,117],"prove":[52],"bijectively":[53],"that":[54,69],"this":[55],"\"ring\"":[56],"is":[57,73],"indeed":[58],"associative.":[59],"This,":[60],"Pieri":[63],"rule,":[64],"give":[65,118],"self-contained":[67],"proof":[68],"ring":[72,81],"isomorphic":[74],"as":[75],"ring-with-basis":[77],"representation":[80],"C})$.":[84],"In":[85],"interpretation,":[88],"becomes":[92],"\"scattering\"":[93],"honeycombs.":[96],"This":[97],"recovers":[98],"some":[99],"\"crosses":[102],"wrenches\"":[104],"diagrams":[105],"Speyer's":[107],"very":[108],"recent":[109],"preprint":[110],"[\"Perfect":[111],"matchings$\\dots$\"],":[112],"whose":[113],"results":[114],"we":[115],"closed":[120],"form":[121],"associativity":[124],"bijection.":[125]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":3},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":3},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}