{"corpus_id":318569,"paper_sha":"ad547fb1ea6ed1c609cc35cb79f4b539fd730929","doi":"10.37236/1860","arxiv_id":"math/0401006","pmid":null,"pmcid":null,"mag_id":1522774938,"dblp_id":"journals/combinatorics/BjornerW04","acl_id":null,"title":"Geometrically Constructed Bases for Homology of Partition Lattices of Types A, B and D","year":2004,"publication_date":"2004-01-02","venue":"Electronic Journal of Combinatorics","journal":{"name":"Electron. J. Comb.","pages":null,"volume":"11"},"journal_issn":null,"journal_title":null,"publication_types":["JournalArticle"],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics","Computer Science"],"reference_count":28,"citation_count":17,"influential_citation_count":1,"is_open_access":true,"arxiv_categories":["math.CO"],"arxiv_license":null,"arxiv_journal_ref":null,"mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i2r3/pdf","s2_open_access_landing_url":"https://www.semanticscholar.org/paper/ad547fb1ea6ed1c609cc35cb79f4b539fd730929","s2_open_access_license":null,"s2_open_access_status":"GOLD","pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"We use the theory of hyperplane arrangements to construct natural bases for the homology of partition lattices of types A, B and D. This extends and explains the “splitting basis” for the homology of the partition lattice given in [20], thus answering a question asked by R. Stanley. More explicitly, the following general technique is presented and utilized. Let A be a central and essential hyperplane arrangement in R d .L etR1 ,...,R k be the bounded regions of a generic hyperplane section of A. We show that there are induced polytopal cycles Ri in the homology of the proper part LA of the intersection lattice such that {Ri }i=1,...,k is a basis for e Hd 2(LA). This geometric method for constructing combinatorial homology bases is applied to the Coxeter arrangements of types A, B and D, and to some interpolating arrangements.","claims":[{"public_id":"cl_10b839a7b1eafe08c29c9278e666f370","status":"active","text":"For a central and essential hyperplane arrangement, the bounded regions of a generic hyperplane section induce polytopal cycles that form a basis for reduced homology of the proper part of the intersection lattice.","confidence":0.97,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_10b839a7b1eafe08c29c9278e666f370"},{"public_id":"cl_305fd51c863ffb3044b2e96a931b0281","status":"active","text":"Natural bases for the homology of partition lattices of types A, B, and D are constructed using hyperplane arrangement theory.","confidence":0.98,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_305fd51c863ffb3044b2e96a931b0281"},{"public_id":"cl_62dbceb7a27314d8dbfa0c9a9edb00ce","status":"active","text":"The construction extends and explains the splitting basis for the homology of the partition lattice, answering a question posed by R. Stanley.","confidence":0.96,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_62dbceb7a27314d8dbfa0c9a9edb00ce"},{"public_id":"cl_85bc8e363b7335146ffd0e9ba0df3843","status":"active","text":"The geometric basis-construction method is applied to the Coxeter arrangements of types A, B, and D and to interpolating arrangements.","confidence":0.94,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_85bc8e363b7335146ffd0e9ba0df3843"}],"concepts":[{"public_id":"co_082847ce7866861178f1a6121d467028","status":"active","name":"partition lattice","description":"The lattice of set partitions whose homology is studied and compared in the cited work.","types":["lattice"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_082847ce7866861178f1a6121d467028"},{"public_id":"co_396360e3f22556fd2c6278cb3cdce777","status":"active","name":"splitting basis","description":"A previously defined basis for the homology of the partition lattice.","types":["basis"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_396360e3f22556fd2c6278cb3cdce777"},{"public_id":"co_46e3d0f10c053e5c57bdbfdfac491682","status":"active","name":"R. Stanley question","description":"The question posed by Richard Stanley that the basis construction is said to answer.","types":["research question"],"aliases":["Stanley question"],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_46e3d0f10c053e5c57bdbfdfac491682"},{"public_id":"co_550580682023ff81d79ef6dd4f264c4d","status":"active","name":"geometric method for constructing combinatorial homology bases","description":"A procedure that uses geometric data from hyperplane arrangements to produce combinatorial homology bases.","types":["method"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_550580682023ff81d79ef6dd4f264c4d"},{"public_id":"co_6d7d6c8e3a1b819f0d2d75aac78a1a7f","status":"active","name":"Coxeter arrangements of types A, B and D","description":"The Coxeter hyperplane arrangements associated with types A, B, and D.","types":["hyperplane arrangement"],"aliases":["Coxeter arrangements"],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_6d7d6c8e3a1b819f0d2d75aac78a1a7f"},{"public_id":"co_9198470b3a7db68f1890fd7c912a58ba","status":"active","name":"partition lattices of types A, B and D","description":"Partition lattices associated with the Coxeter types A, B, and D.","types":["poset","lattice"],"aliases":["partition lattice of type A","partition lattice of type B","partition lattice of type D"],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_9198470b3a7db68f1890fd7c912a58ba"},{"public_id":"co_aa166aff479f354a1405447bf367758c","status":"active","name":"hyperplane arrangements","description":"Collections of hyperplanes used here to build homology bases for intersection lattices.","types":["method","geometric object"],"aliases":["arrangements of hyperplanes"],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_aa166aff479f354a1405447bf367758c"},{"public_id":"co_b774675a4236d8301ac3322034d84910","status":"active","name":"polytopal cycles","description":"Cycles arising from bounded regions that represent homology classes.","types":["cycle"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_b774675a4236d8301ac3322034d84910"},{"public_id":"co_bb86929de4bcdc13c9d8b2e942b5407e","status":"active","name":"interpolating arrangements","description":"Arrangements that interpolate between the Coxeter-type cases considered in the paper.","types":["hyperplane arrangement"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_bb86929de4bcdc13c9d8b2e942b5407e"},{"public_id":"co_d16472f6d5e5c8e849b10dfebc3fd870","status":"active","name":"generic hyperplane section","description":"A hyperplane slice chosen generically from the arrangement.","types":["geometric construction"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_d16472f6d5e5c8e849b10dfebc3fd870"},{"public_id":"co_e2cc504467127ec487ae245261c09770","status":"active","name":"central and essential hyperplane arrangement","description":"A hyperplane arrangement with all hyperplanes passing through the origin and with full-rank intersection.","types":["geometric object"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_e2cc504467127ec487ae245261c09770"},{"public_id":"co_e83e079b0d08c5358043cd0b03b130f1","status":"active","name":"homology","description":"An algebraic-topological invariant of the relevant lattices and simplicial complexes.","types":["topological invariant"],"aliases":[],"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/concepts/co_e83e079b0d08c5358043cd0b03b130f1"}],"external_ids":{"DOI":"10.37236/1860","ArXiv":"math/0401006","PubMed":null,"PubMedCentral":null,"MAG":1522774938,"DBLP":"journals/combinatorics/BjornerW04","ACL":null},"open_access":{"is_open_access":true,"pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i2r3/pdf","landing_url":"https://www.semanticscholar.org/paper/ad547fb1ea6ed1c609cc35cb79f4b539fd730929","source":"semantic_scholar","pdf_url_source":"semantic_scholar_open_access_pdf","license":null,"status":"GOLD","reason":null},"reference_availability":{"status":"available","references_indexed":true,"full_text_available":true,"full_text_source":"arxiv","count_basis":"semantic_scholar_metadata","extraction_status":"not_applicable","reason":null},"source":{"provider":"episteme2","base_corpus":"semantic_scholar_dump","freshness_mode":"unknown","basis":["semantic_scholar_metadata","postgres_metadata"],"limits":["paper metadata is based on indexed upstream scholarly datasets","claims and concepts are available only for extracted papers","absence of claims or concepts means no extracted graph data is available in this response"],"status":"available","degraded":false,"degraded_reasons":[],"diagnostics":{"status":"available","degraded":false,"degraded_reasons":[],"metadata_status":"available","graph_status":"available","abstract_status":"available"},"source_flags":1},"paper_id":635162,"paper_uid":"b3cb5a83-036e-4066-8920-81ad9e017a74","canonical_identity":{"paper_id":635162,"paper_uid":"b3cb5a83-036e-4066-8920-81ad9e017a74","identity_status":"available","lookup_basis":"semantic_scholar_external_id","compatibility_path":"corpus_id"},"url":"https://sah.borca.ai/papers/318569"}