{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:4LUTL4GK2R6LCG433PPU2L32UN","short_pith_number":"pith:4LUTL4GK","schema_version":"1.0","canonical_sha256":"e2e935f0cad47cb11b9bdbdf4d2f7aa37281511d7f2daadbf8521bba145f5eab","source":{"kind":"arxiv","id":"1608.07918","version":3},"attestation_state":"computed","paper":{"title":"Minimal right determiners of irreducible morphisms in algebras of type ${\\mathbb A}_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Xiaoxing Wu, Zhaoyong Huang","submitted_at":"2016-08-29T06:07:28Z","abstract_excerpt":"Let $\\Lambda$ be a finite dimensional algebra of type ${\\mathbb A}_n$ over an algebraically closed field $K$ with the quiver $Q$ and let $|\\Det(\\Lambda)|$ be the number of the minimal right determiners of all irreducible morphisms between indecomposable left $\\Lambda$-modules. If $\\Lambda$ is a path algebra, then we have $$|\\Det(\\Lambda)|= 2n-2, &\\mbox{if $p=0$; } 2n-p-1, &\\mbox{if $p\\geq 1$,}$$ where $p=|\\{i\\mid i$ is a source in $Q$ with $2\\leq i\\leq n-1\\}|$. If $\\Lambda$ is a bound quiver algebra, then we have $$ |\\Det(\\Lambda)|= 2n-2, &\\mbox{if $r=1$; } 2n-p-q-1, &\\mbox{if $r\\geq 2$,} $$ w"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1608.07918","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-08-29T06:07:28Z","cross_cats_sorted":[],"title_canon_sha256":"ab05345ce5c71aa373d4b9e0c2ff2a7b3d534e652a5dafdf21eba2df11d16f68","abstract_canon_sha256":"6f006277cec4ee4dfa0cff628e8ec8225b41f83100a6ad47fceb765a445d3787"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:42.419622Z","signature_b64":"q1safRd7fsaevyCoSnp0kkxG0N3P4lNdPu7h6Aokmi/W4aljNfyF7aTpcN72QF1clmwa6vK5rdgTf+f3Fl1uBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e2e935f0cad47cb11b9bdbdf4d2f7aa37281511d7f2daadbf8521bba145f5eab","last_reissued_at":"2026-05-18T00:48:42.419101Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:42.419101Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimal right determiners of irreducible morphisms in algebras of type ${\\mathbb A}_n$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Xiaoxing Wu, Zhaoyong Huang","submitted_at":"2016-08-29T06:07:28Z","abstract_excerpt":"Let $\\Lambda$ be a finite dimensional algebra of type ${\\mathbb A}_n$ over an algebraically closed field $K$ with the quiver $Q$ and let $|\\Det(\\Lambda)|$ be the number of the minimal right determiners of all irreducible morphisms between indecomposable left $\\Lambda$-modules. If $\\Lambda$ is a path algebra, then we have $$|\\Det(\\Lambda)|= 2n-2, &\\mbox{if $p=0$; } 2n-p-1, &\\mbox{if $p\\geq 1$,}$$ where $p=|\\{i\\mid i$ is a source in $Q$ with $2\\leq i\\leq n-1\\}|$. If $\\Lambda$ is a bound quiver algebra, then we have $$ |\\Det(\\Lambda)|= 2n-2, &\\mbox{if $r=1$; } 2n-p-q-1, &\\mbox{if $r\\geq 2$,} $$ w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.07918","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1608.07918","created_at":"2026-05-18T00:48:42.419174+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.07918v3","created_at":"2026-05-18T00:48:42.419174+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.07918","created_at":"2026-05-18T00:48:42.419174+00:00"},{"alias_kind":"pith_short_12","alias_value":"4LUTL4GK2R6L","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_16","alias_value":"4LUTL4GK2R6LCG43","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_8","alias_value":"4LUTL4GK","created_at":"2026-05-18T12:29:58.707656+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4LUTL4GK2R6LCG433PPU2L32UN","json":"https://pith.science/pith/4LUTL4GK2R6LCG433PPU2L32UN.json","graph_json":"https://pith.science/api/pith-number/4LUTL4GK2R6LCG433PPU2L32UN/graph.json","events_json":"https://pith.science/api/pith-number/4LUTL4GK2R6LCG433PPU2L32UN/events.json","paper":"https://pith.science/paper/4LUTL4GK"},"agent_actions":{"view_html":"https://pith.science/pith/4LUTL4GK2R6LCG433PPU2L32UN","download_json":"https://pith.science/pith/4LUTL4GK2R6LCG433PPU2L32UN.json","view_paper":"https://pith.science/paper/4LUTL4GK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.07918&json=true","fetch_graph":"https://pith.science/api/pith-number/4LUTL4GK2R6LCG433PPU2L32UN/graph.json","fetch_events":"https://pith.science/api/pith-number/4LUTL4GK2R6LCG433PPU2L32UN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4LUTL4GK2R6LCG433PPU2L32UN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4LUTL4GK2R6LCG433PPU2L32UN/action/storage_attestation","attest_author":"https://pith.science/pith/4LUTL4GK2R6LCG433PPU2L32UN/action/author_attestation","sign_citation":"https://pith.science/pith/4LUTL4GK2R6LCG433PPU2L32UN/action/citation_signature","submit_replication":"https://pith.science/pith/4LUTL4GK2R6LCG433PPU2L32UN/action/replication_record"}},"created_at":"2026-05-18T00:48:42.419174+00:00","updated_at":"2026-05-18T00:48:42.419174+00:00"}