{"id":"https://openalex.org/W2609703094","doi":null,"title":"On an involution on the set of Littlewood-Richardson tableaux and the hidden commutativity","display_name":"On an involution on the set of Littlewood-Richardson tableaux and the hidden commutativity","publication_year":2000,"publication_date":"2000-01-01","ids":{"openalex":"https://openalex.org/W2609703094","mag":"2609703094"},"language":"en","primary_location":{"id":"pmh:oai:estudogeral.sib.uc.pt:10316/11472","doi":null,"is_oa":true,"landing_page_url":"http://hdl.handle.net/10316/11472","pdf_url":"https://estudogeral.uc.pt/bitstream/10316/11472/1/On%20an%20involution%20on%20the%20set%20of%20Littlewood-Richardson%20tableaux%20and%20the%20hidden%20commutativity.pdf","source":{"id":"https://openalex.org/S4306402433","display_name":"Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"http://purl.org/coar/resource_type/c_816b"},"volume":null,"type":"preprint","indexed_in":[],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://estudogeral.uc.pt/bitstream/10316/11472/1/On%20an%20involution%20on%20the%20set%20of%20Littlewood-Richardson%20tableaux%20and%20the%20hidden%20commutativity.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5057861201","display_name":"Olga Azenhas","orcid":"https://orcid.org/0000-0001-7718-7158","observed_orcids":["https://orcid.org/0000-0001-7718-7158"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Olga Azenhas","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0001-7718-7158","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5057861201"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"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":1.0,"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":1.0,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.0,"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"}},{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.0,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/littlewood-richardson-rule","display_name":"Littlewood-Richardson rule","score":0.9549999833106995},{"id":"https://openalex.org/keywords/involutions","display_name":"involutions","score":0.8230000138282776},{"id":"https://openalex.org/keywords/littlewood-richardson-coefficients","display_name":"Littlewood-Richardson coefficients","score":0.9789999723434448},{"id":"https://openalex.org/keywords/nested-decomposition","display_name":"nested decomposition","score":0.3440000116825104}],"concepts":[{"id":"https://openalex.org/C2781409172","wikidata":"https://www.wikidata.org/wiki/Q17056802","display_name":"Involution (esoterism)","level":3,"score":0.6987794637680054},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.6245057582855225},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5652178525924683},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4186762571334839},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3297910690307617},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.13705068826675415},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.1286674439907074},{"id":"https://openalex.org/C186720457","wikidata":"https://www.wikidata.org/wiki/Q7087","display_name":"Consciousness","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"pmh:oai:estudogeral.sib.uc.pt:10316/11472","doi":null,"is_oa":true,"landing_page_url":"http://hdl.handle.net/10316/11472","pdf_url":"https://estudogeral.uc.pt/bitstream/10316/11472/1/On%20an%20involution%20on%20the%20set%20of%20Littlewood-Richardson%20tableaux%20and%20the%20hidden%20commutativity.pdf","source":{"id":"https://openalex.org/S4306402433","display_name":"Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"http://purl.org/coar/resource_type/c_816b"},{"id":"pmh:oai:estudogeral.uc.pt:10316/11472","doi":null,"is_oa":true,"landing_page_url":"https://hdl.handle.net/10316/11472","pdf_url":null,"source":{"id":"https://openalex.org/S4306402433","display_name":"Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"preprint"},{"id":"mag:2609703094","doi":null,"is_oa":false,"landing_page_url":"https://estudogeral.sib.uc.pt/bitstream/10316/11472/1/On%20an%20involution%20on%20the%20set%20of%20Littlewood-Richardson%20tableaux%20and%20the%20hidden%20commutativity.pdf","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:estudogeral.sib.uc.pt:10316/11472","doi":null,"is_oa":true,"landing_page_url":"http://hdl.handle.net/10316/11472","pdf_url":"https://estudogeral.uc.pt/bitstream/10316/11472/1/On%20an%20involution%20on%20the%20set%20of%20Littlewood-Richardson%20tableaux%20and%20the%20hidden%20commutativity.pdf","source":{"id":"https://openalex.org/S4306402433","display_name":"Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"http://purl.org/coar/resource_type/c_816b"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2609703094.pdf","grobid_xml":"https://content.openalex.org/works/W2609703094.grobid-xml"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W1500907847","https://openalex.org/W1607638543","https://openalex.org/W1664429083","https://openalex.org/W1975605482","https://openalex.org/W2000206862","https://openalex.org/W2010276484","https://openalex.org/W2019550287","https://openalex.org/W2024063662","https://openalex.org/W2028732254","https://openalex.org/W2049157850","https://openalex.org/W2063173931","https://openalex.org/W2087792692","https://openalex.org/W2132372851","https://openalex.org/W2134984950"],"related_works":["https://openalex.org/W2297385013","https://openalex.org/W1997526206","https://openalex.org/W2122032220","https://openalex.org/W2904534706","https://openalex.org/W2036568559","https://openalex.org/W2151016043","https://openalex.org/W4414889879","https://openalex.org/W6961138501","https://openalex.org/W7140848769","https://openalex.org/W2143851795"],"abstract_inverted_index":{"The":[0,25,83],"original":[1,212],"definition":[2,61],"of":[3,35,43,50,58,66,69,74,86,92,96,108,111,137,144,148,157,160,205,210,218,221,245,248,263,266,282],"the":[4,16,22,33,56,59,90,193,206,211,219,222,227,238,242,261,273,283,285],"Littlewood-Richardson":[5,23,223],"(LR)":[6],"rule":[7,240,257,287],"for":[8,20,79],"composing":[9],"partitions":[10,139],"is":[11,28,63,89,134,179,235,258],"exclusively":[12],"considered,":[13],"i.":[14],"e.,":[15],"classical":[17],"combinatorial":[18,183,203],"device":[19],"calculating":[21],"coefficients.":[24],"main":[26,84],"result":[27],"an":[29,40,93,145],"explicit":[30],"involution":[31,60,197,275],"on":[32,188],"set":[34],"LR":[36,41,67,94,109,146,158,190,194,213,239,286],"tableaux":[37,68,110,159],"which":[38,140],"transforms":[39],"tableau":[42,95,147,191],"type":[44,51,100,115,149,164],"[a,":[45,101],"b,":[46,102],"c]":[47,103,152],"into":[48,72,104,153],"one":[49],"[b,":[52,150],"a,":[53,151],"c].":[54],"On":[55],"basis":[57],"it":[62],"a":[64,105,135,142,154,182,189,199,246,264,279],"projection":[65,88,178],"order":[70,75,97,112,161],"r":[71,76,80,98],"those":[73],"-":[77],"1,":[78,128,174],"~":[81],"1.":[82],"feature":[85],"this":[87,256],"decomposition":[91,143],"and":[99,114,163,185,201,288],"nested":[106,155],"sequence":[107,136,156],"s":[113,126,162,172],"[a(s),":[116],"(b1,":[117],"...":[118,122,129,166,169,175],",":[119,123,130],"bs);":[120],"(Cr-s+l,":[121,168],"cr":[124],")],":[125,171],"==":[127,173,230],"r,":[131],"where":[132],"(a(s))~==l":[133],"interlacing":[138],"defines":[141],"[(b1,":[165],",bs);a(s);":[167],",cr":[170],",r.":[176],"This":[177,196],"accomplished":[180],"introducing":[181],"deletion":[184],"insertion":[186],"operation":[187],"preserving":[192],"conditions.":[195],"yields":[198],"self-contained":[200],"direct":[202],"interpretation":[204],"well-known":[207],"commutative":[208],"property":[209],"rule,":[214],"as":[215,217],"well":[216],"symmetry":[220],"coefficients":[224],"given":[225],"by":[226],"equality":[228],"Ngb":[229],"N":[231],"ba":[232],".":[233],"It":[234,251],"known":[236],"that":[237,255],"describes":[241],"Smith":[243],"invariants":[244],"product":[247],"integral":[249],"matrices.":[250],"has":[252],"been":[253],"proven":[254],"also":[259],"describing":[260],"eigenvalues":[262],"sum":[265],"Hermitian":[267],"matrices":[268],"[13,":[269],"14,":[270],"17].":[271],"With":[272],"present":[274],"we":[276],"aim":[277],"to":[278],"deeper":[280],"understanding":[281],"structure":[284],"its":[289],"relationship":[290],"with":[291],"these":[292],"two":[293],"problems":[294],"in":[295],"matrix":[296],"theory.":[297]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2017-05-05T00:00:00"}