{"id":"https://openalex.org/W2758566348","doi":"https://doi.org/10.37236/6960","title":"Cohomology Classes of Interval Positroid Varieties and a Conjecture of Liu","display_name":"Cohomology Classes of Interval Positroid Varieties and a Conjecture of Liu","publication_year":2018,"publication_date":"2018-10-05","ids":{"openalex":"https://openalex.org/W2758566348","doi":"https://doi.org/10.37236/6960","mag":"2758566348"},"language":"en","primary_location":{"id":"doi:10.37236/6960","is_oa":true,"landing_page_url":"https://doi.org/10.37236/6960","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v25i4p4/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v25i4p4/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5087936199","display_name":"Brendan Pawlowski","orcid":"https://orcid.org/0000-0003-0681-813X"},"institutions":[{"id":"https://openalex.org/I2800817003","display_name":"Southern California University for Professional Studies","ror":"https://ror.org/058zz0t50","country_code":"US","type":"education","lineage":["https://openalex.org/I2800817003"]},{"id":"https://openalex.org/I1174212","display_name":"University of Southern California","ror":"https://ror.org/03taz7m60","country_code":"US","type":"education","lineage":["https://openalex.org/I1174212"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Brendan Pawlowski","raw_affiliation_strings":["University of Southern California","University  of Southern California"],"affiliations":[{"raw_affiliation_string":"University of Southern California","institution_ids":["https://openalex.org/I2800817003","https://openalex.org/I1174212"]},{"raw_affiliation_string":"University  of Southern California","institution_ids":["https://openalex.org/I2800817003","https://openalex.org/I1174212"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5087936199"],"corresponding_institution_ids":["https://openalex.org/I1174212","https://openalex.org/I2800817003"],"apc_list":null,"apc_paid":null,"fwci":0.5595,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.56214866,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"25","issue":"4","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":0.9991999864578247,"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":0.9991999864578247,"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":0.9973000288009644,"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9837999939918518,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8045182824134827},{"id":"https://openalex.org/keywords/subvariety","display_name":"Subvariety","score":0.7481889724731445},{"id":"https://openalex.org/keywords/cohomology","display_name":"Cohomology","score":0.6194753050804138},{"id":"https://openalex.org/keywords/diagram","display_name":"Diagram","score":0.6010850667953491},{"id":"https://openalex.org/keywords/schubert-variety","display_name":"Schubert variety","score":0.5280638337135315},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.5211279988288879},{"id":"https://openalex.org/keywords/grassmannian","display_name":"Grassmannian","score":0.5123645067214966},{"id":"https://openalex.org/keywords/variety","display_name":"Variety (cybernetics)","score":0.5002899169921875},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.48664960265159607},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.48206859827041626},{"id":"https://openalex.org/keywords/schubert-calculus","display_name":"Schubert calculus","score":0.4543260335922241},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.4394393563270569},{"id":"https://openalex.org/keywords/quiver","display_name":"Quiver","score":0.43427103757858276}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8045182824134827},{"id":"https://openalex.org/C2777183706","wikidata":"https://www.wikidata.org/wiki/Q630771","display_name":"Subvariety","level":3,"score":0.7481889724731445},{"id":"https://openalex.org/C78606066","wikidata":"https://www.wikidata.org/wiki/Q1198376","display_name":"Cohomology","level":2,"score":0.6194753050804138},{"id":"https://openalex.org/C186399060","wikidata":"https://www.wikidata.org/wiki/Q959962","display_name":"Diagram","level":2,"score":0.6010850667953491},{"id":"https://openalex.org/C47158899","wikidata":"https://www.wikidata.org/wiki/Q7432938","display_name":"Schubert variety","level":2,"score":0.5280638337135315},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5211279988288879},{"id":"https://openalex.org/C162929932","wikidata":"https://www.wikidata.org/wiki/Q129638","display_name":"Grassmannian","level":2,"score":0.5123645067214966},{"id":"https://openalex.org/C136197465","wikidata":"https://www.wikidata.org/wiki/Q1729295","display_name":"Variety (cybernetics)","level":2,"score":0.5002899169921875},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.48664960265159607},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.48206859827041626},{"id":"https://openalex.org/C169843257","wikidata":"https://www.wikidata.org/wiki/Q7432936","display_name":"Schubert calculus","level":3,"score":0.4543260335922241},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.4394393563270569},{"id":"https://openalex.org/C168310172","wikidata":"https://www.wikidata.org/wiki/Q493980","display_name":"Quiver","level":2,"score":0.43427103757858276},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.37236/6960","is_oa":true,"landing_page_url":"https://doi.org/10.37236/6960","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v25i4p4/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1410.7419","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1410.7419","pdf_url":"https://arxiv.org/pdf/1410.7419","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"mag:2758566348","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1410.7419.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"doi:10.48550/arxiv.1410.7419","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1410.7419","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.37236/6960","is_oa":true,"landing_page_url":"https://doi.org/10.37236/6960","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v25i4p4/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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Reduced inequalities","score":0.4399999976158142,"id":"https://metadata.un.org/sdg/10"}],"awards":[{"id":"https://openalex.org/G5387366962","display_name":"Combinatorial and Algebraic Aspects of Varieties","funder_award_id":"1101017","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G63536932","display_name":null,"funder_award_id":"DMS-1101017","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2758566348.pdf","grobid_xml":"https://content.openalex.org/works/W2758566348.grobid-xml"},"referenced_works_count":16,"referenced_works":["https://openalex.org/W1496283617","https://openalex.org/W1518468345","https://openalex.org/W1553824818","https://openalex.org/W1974449201","https://openalex.org/W1975418884","https://openalex.org/W1995154714","https://openalex.org/W2011208855","https://openalex.org/W2053622294","https://openalex.org/W2060713787","https://openalex.org/W2066798586","https://openalex.org/W2073585131","https://openalex.org/W2073876300","https://openalex.org/W2076451143","https://openalex.org/W2109001550","https://openalex.org/W2139572697","https://openalex.org/W6766135204"],"related_works":["https://openalex.org/W2964110644","https://openalex.org/W1689364629","https://openalex.org/W2604016210","https://openalex.org/W2964082626","https://openalex.org/W2916547361","https://openalex.org/W3122983934","https://openalex.org/W2951575286","https://openalex.org/W2073980466","https://openalex.org/W3065843010","https://openalex.org/W2962870433","https://openalex.org/W2951268769","https://openalex.org/W2022304678","https://openalex.org/W2972666529","https://openalex.org/W2102075924","https://openalex.org/W2951387589","https://openalex.org/W107244006","https://openalex.org/W2901913779","https://openalex.org/W1934138896","https://openalex.org/W3103836366","https://openalex.org/W1965810222"],"abstract_inverted_index":{"To":[0,101],"each":[1],"finite":[2],"subset":[3],"of":[4,13,23,37,47,66,98,116,118,132,161],"$\\mathbb{Z}^2$":[5],"(a":[6,17,27],"diagram),":[7],"one":[8,117,157],"can":[9],"associate":[10],"a":[11,14,21,24,38,54,67,94,111,114,146],"subvariety":[12],"complex":[15],"Grassmannian":[16],"diagram":[18,39,64,108],"variety),":[19],"and":[20,164],"representation":[22],"symmetric":[25,142,155],"group":[26],"Specht":[28,50],"module).":[29],"Liu":[30],"has":[31],"conjectured":[32,71],"that":[33,61,89],"the":[34,44,48,63,82,87,106,129,159],"cohomology":[35,72,130],"class":[36,73,84],"variety":[40,65,109],"is":[41,75,93],"represented":[42,138],"by":[43,139],"Frobenius":[45],"characteristic":[46],"corresponding":[49],"module.":[51],"We":[52,144],"give":[53,145],"counterexample":[55],"to":[56,124],"this":[57],"conjecture.However,":[58],"we":[59,104],"show":[60],"for":[62,149],"permutation":[68],"diagram,":[69],"Liu's":[70],"$\\sigma$":[74],"at":[76],"least":[77],"an":[78],"upper":[79],"bound":[80],"on":[81],"actual":[83],"$\\tau$,":[85],"in":[86,113],"sense":[88],"$\\sigma":[90],"-":[91],"\\tau$":[92],"nonnegative":[95],"linear":[96],"combination":[97],"Schubert":[99],"classes.":[100],"do":[102],"this,":[103],"exhibit":[105],"appropriate":[107],"as":[110,152],"component":[112],"degeneration":[115],"Knutson's":[119],"interval":[120,134],"positroid":[121,135],"varieties":[122,136],"(up":[123],"Grassmann":[125],"duality).":[126],"A":[127],"priori,":[128],"classes":[131,151],"these":[133,150],"are":[137],"affine":[140],"Stanley":[141,154],"functions.":[143],"different":[147],"formula":[148],"ordinary":[153],"functions,":[156],"with":[158,166],"advantage":[160],"being":[162],"Schur-positive":[163],"compatible":[165],"inclusions":[167],"between":[168],"Grassmannians.":[169]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}