{"id":"https://openalex.org/W7213366663","doi":"https://doi.org/10.48550/arxiv.2609.14240","title":"Graded Kronecker Modules of Constant Rank","display_name":"Graded Kronecker Modules of Constant Rank","publication_year":2026,"publication_date":"2026-09-13","ids":{"openalex":"https://openalex.org/W7213366663","doi":"https://doi.org/10.48550/arxiv.2609.14240"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.14240","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14240","pdf_url":"https://arxiv.org/pdf/2609.14240","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.14240","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151832348","display_name":"Jie Liu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Liu, Jie","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5151832348"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9980000257492065,"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"}},"topics":[{"id":"https://openalex.org/T10287","display_name":"Algebraic structures and combinatorial models","score":0.9980000257492065,"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.0006000000284984708,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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":0.00019999999494757503,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/category-of-representations","display_name":"category of representations","score":0.33500000834465027}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8270999789237976},{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.7944999933242798},{"id":"https://openalex.org/C168310172","wikidata":"https://www.wikidata.org/wiki/Q493980","display_name":"Quiver","level":2,"score":0.7296000123023987},{"id":"https://openalex.org/C128107574","wikidata":"https://www.wikidata.org/wiki/Q182003","display_name":"Injective function","level":2,"score":0.7129999995231628},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.616100013256073},{"id":"https://openalex.org/C119238805","wikidata":"https://www.wikidata.org/wiki/Q229102","display_name":"Surjective function","level":2,"score":0.5882999897003174},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.5737000107765198},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5709999799728394},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.4997999966144562},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4404999911785126},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.43220001459121704},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.40450000762939453},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.39489999413490295},{"id":"https://openalex.org/C99463465","wikidata":"https://www.wikidata.org/wiki/Q2716519","display_name":"Injective module","level":3,"score":0.39410001039505005},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.38580000400543213},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.35019999742507935},{"id":"https://openalex.org/C100036529","wikidata":"https://www.wikidata.org/wiki/Q1945067","display_name":"Monomorphism","level":3,"score":0.30300000309944153},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.28529998660087585},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.28029999136924744},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.26820001006126404},{"id":"https://openalex.org/C118712358","wikidata":"https://www.wikidata.org/wiki/Q782566","display_name":"Automorphism","level":2,"score":0.26030001044273376},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.26019999384880066}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.14240","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14240","pdf_url":"https://arxiv.org/pdf/2609.14240","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"doi:10.48550/arxiv.2609.14240","doi":"https://doi.org/10.48550/arxiv.2609.14240","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.14240","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.14240","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14240","pdf_url":"https://arxiv.org/pdf/2609.14240","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213366663.pdf","grobid_xml":"https://content.openalex.org/works/W7213366663.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W2964030154","https://openalex.org/W7171503443"],"related_works":["https://openalex.org/W7039531082","https://openalex.org/W2750971164","https://openalex.org/W2016442784","https://openalex.org/W4392425844","https://openalex.org/W589517418","https://openalex.org/W2587944150","https://openalex.org/W2416859244","https://openalex.org/W7213322278","https://openalex.org/W2769043707","https://openalex.org/W2341575154"],"abstract_inverted_index":{"Let":[0],"$K(n)$":[1],"be":[2,10],"the":[3,11,64],"generalized":[4],"Kronecker":[5],"quiver,":[6],"and":[7,19,50,60,71],"let":[8],"$(T(n),\u03a9)$":[9,34],"universal":[12],"covering":[13],"of":[14,30,53],"$K(n)$,":[15],"where":[16],"$n\\geq":[17],"3$":[18],"$\u03a9$":[20],"is":[21],"a":[22,41,48],"bipartite":[23],"orientation.":[24],"We":[25,56],"introduce":[26],"three":[27],"different":[28],"types":[29],"representations":[31,51],"for":[32],"quiver":[33],"by":[35],"$K(n)$:":[36],"maps":[37,43],"are":[38,44,52],"injective":[39],"in":[40,47],"representation":[42],"all":[45],"surjective":[46],"representation,":[49],"constant":[54],"rank.":[55],"use":[57],"$\\Inj,":[58],"\\Sur$":[59],"$\\GR$":[61],"to":[62],"denote":[63],"corresponding":[65],"subcategories":[66],"containing":[67],"these":[68],"representations,":[69],"respectively,":[70],"we":[72],"show":[73],"that":[74],"\\begin{center}":[75],"$\\GR=":[76],"\\Inj\\cup\\Sur$.":[77],"\\end{center}":[78]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-16T00:00:00"}