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We prove that ${\\rm rep.dim}\\ T_2(A)$ is at most three if $A$ is Dynkin type and ${\\rm rep.dim}\\ T_2(A)$ is at most four if $A$ is not Dynkin type. Let $T$ be a tilting A-$\\module$ and $\\ol{T}=T\\oplus\\ol{P}$ be a tilting $A^{(1)}$-$\\module$. We show that $\\End_{A^{(1)}} \\ol{T}$ is representation finite if and only if the full s"},"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":"1107.3865","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-07-19T23:59:03Z","cross_cats_sorted":[],"title_canon_sha256":"b4aaa2df499b42320293f851463fbd982745e8524c4867ac4e8a6e19e7702608","abstract_canon_sha256":"4de51aee0f8db42874fde466546a6965628c7cd49b9a1cf2de88c30f47995f3d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:41.362158Z","signature_b64":"ggnDkB1AdI/6gTVsmPRtBdM3P01KbXskbl0//pHaYl+cC3gMRVFH0vIuppRPbKNDNFYvGc/tZoJ8Q9VkuZKTCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"41225eddc82a11d1f84487cdc0bd00bc4656325bf8cbf05673a7db947f4b2141","last_reissued_at":"2026-05-18T03:35:41.361276Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:41.361276Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Representation dimensions of triangular matrix algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Hongbo Yin, Shunhua Zhang","submitted_at":"2011-07-19T23:59:03Z","abstract_excerpt":"Let $A$ be a finite dimensional hereditary algebra over an algebraically closed field $k$, $T_2(A)=(\\begin{array}{cc}A&0 A&A\\end{array})$ be the triangular matrix algebra and $A^{(1)}=(\\begin{array}{cc}A&0 DA&A\\end{array})$ be the duplicated algebra of $A$ respectively. We prove that ${\\rm rep.dim}\\ T_2(A)$ is at most three if $A$ is Dynkin type and ${\\rm rep.dim}\\ T_2(A)$ is at most four if $A$ is not Dynkin type. Let $T$ be a tilting A-$\\module$ and $\\ol{T}=T\\oplus\\ol{P}$ be a tilting $A^{(1)}$-$\\module$. We show that $\\End_{A^{(1)}} \\ol{T}$ is representation finite if and only if the full s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.3865","kind":"arxiv","version":1},"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":"1107.3865","created_at":"2026-05-18T03:35:41.361447+00:00"},{"alias_kind":"arxiv_version","alias_value":"1107.3865v1","created_at":"2026-05-18T03:35:41.361447+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.3865","created_at":"2026-05-18T03:35:41.361447+00:00"},{"alias_kind":"pith_short_12","alias_value":"IERF5XOIFII5","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_16","alias_value":"IERF5XOIFII5D6CE","created_at":"2026-05-18T12:26:30.835961+00:00"},{"alias_kind":"pith_short_8","alias_value":"IERF5XOI","created_at":"2026-05-18T12:26:30.835961+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/IERF5XOIFII5D6CEQ7G4BPIAXR","json":"https://pith.science/pith/IERF5XOIFII5D6CEQ7G4BPIAXR.json","graph_json":"https://pith.science/api/pith-number/IERF5XOIFII5D6CEQ7G4BPIAXR/graph.json","events_json":"https://pith.science/api/pith-number/IERF5XOIFII5D6CEQ7G4BPIAXR/events.json","paper":"https://pith.science/paper/IERF5XOI"},"agent_actions":{"view_html":"https://pith.science/pith/IERF5XOIFII5D6CEQ7G4BPIAXR","download_json":"https://pith.science/pith/IERF5XOIFII5D6CEQ7G4BPIAXR.json","view_paper":"https://pith.science/paper/IERF5XOI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1107.3865&json=true","fetch_graph":"https://pith.science/api/pith-number/IERF5XOIFII5D6CEQ7G4BPIAXR/graph.json","fetch_events":"https://pith.science/api/pith-number/IERF5XOIFII5D6CEQ7G4BPIAXR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IERF5XOIFII5D6CEQ7G4BPIAXR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IERF5XOIFII5D6CEQ7G4BPIAXR/action/storage_attestation","attest_author":"https://pith.science/pith/IERF5XOIFII5D6CEQ7G4BPIAXR/action/author_attestation","sign_citation":"https://pith.science/pith/IERF5XOIFII5D6CEQ7G4BPIAXR/action/citation_signature","submit_replication":"https://pith.science/pith/IERF5XOIFII5D6CEQ7G4BPIAXR/action/replication_record"}},"created_at":"2026-05-18T03:35:41.361447+00:00","updated_at":"2026-05-18T03:35:41.361447+00:00"}