{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:INEJKZ4VHRWCNLEFY7LXKFEPFT","short_pith_number":"pith:INEJKZ4V","schema_version":"1.0","canonical_sha256":"43489567953c6c26ac85c7d775148f2ccea26c2d624a1d6f622ab458cd678a8c","source":{"kind":"arxiv","id":"2203.06238","version":3},"attestation_state":"computed","paper":{"title":"When does the Auslander-Reiten translation operate linearly on the Grothendieck group? -- Part I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Carlo Klapproth","submitted_at":"2022-03-11T20:44:29Z","abstract_excerpt":"For a hereditary, finite-dimensional algebra $A$ the Coxeter transformation extends the action of the Auslander--Reiten translation on the non-projective indecomposable modules to a linear endomorphism of the Grothendieck group of the category of finitely generated $A$-modules. It is natural to ask whether other algebras admit a similar linear extension. We show that this is indeed the case for all Nakayama algebras. Conversely, we show that finite-dimensional algebras with non-acyclic and connected quiver admitting such a linear extension are already cyclic Nakayama algebras."},"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":"2203.06238","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2022-03-11T20:44:29Z","cross_cats_sorted":[],"title_canon_sha256":"6e14f90af412517f01e30be2e3151ae192d8ade318555e2f92e226c8660acfa7","abstract_canon_sha256":"a80423036c4df8e8bf1a54933747cf0a3f7326631e4a211ff5ca5ec1287b22e1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:39.923464Z","signature_b64":"PbDh2F8zUX3Vx9nDdaFeP3IvEn58+Dh13mSUXTLla2cD0/IV2INdKfe77kjXu6u4P0SHRyGqcwmbL0pP2nBiCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"43489567953c6c26ac85c7d775148f2ccea26c2d624a1d6f622ab458cd678a8c","last_reissued_at":"2026-05-18T02:44:39.922926Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:39.922926Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"When does the Auslander-Reiten translation operate linearly on the Grothendieck group? -- Part I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Carlo Klapproth","submitted_at":"2022-03-11T20:44:29Z","abstract_excerpt":"For a hereditary, finite-dimensional algebra $A$ the Coxeter transformation extends the action of the Auslander--Reiten translation on the non-projective indecomposable modules to a linear endomorphism of the Grothendieck group of the category of finitely generated $A$-modules. It is natural to ask whether other algebras admit a similar linear extension. We show that this is indeed the case for all Nakayama algebras. Conversely, we show that finite-dimensional algebras with non-acyclic and connected quiver admitting such a linear extension are already cyclic Nakayama algebras."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2203.06238","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":"2203.06238","created_at":"2026-05-18T02:44:39.923012+00:00"},{"alias_kind":"arxiv_version","alias_value":"2203.06238v3","created_at":"2026-05-18T02:44:39.923012+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2203.06238","created_at":"2026-05-18T02:44:39.923012+00:00"},{"alias_kind":"pith_short_12","alias_value":"INEJKZ4VHRWC","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_16","alias_value":"INEJKZ4VHRWCNLEF","created_at":"2026-05-18T12:33:33.725879+00:00"},{"alias_kind":"pith_short_8","alias_value":"INEJKZ4V","created_at":"2026-05-18T12:33:33.725879+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/INEJKZ4VHRWCNLEFY7LXKFEPFT","json":"https://pith.science/pith/INEJKZ4VHRWCNLEFY7LXKFEPFT.json","graph_json":"https://pith.science/api/pith-number/INEJKZ4VHRWCNLEFY7LXKFEPFT/graph.json","events_json":"https://pith.science/api/pith-number/INEJKZ4VHRWCNLEFY7LXKFEPFT/events.json","paper":"https://pith.science/paper/INEJKZ4V"},"agent_actions":{"view_html":"https://pith.science/pith/INEJKZ4VHRWCNLEFY7LXKFEPFT","download_json":"https://pith.science/pith/INEJKZ4VHRWCNLEFY7LXKFEPFT.json","view_paper":"https://pith.science/paper/INEJKZ4V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2203.06238&json=true","fetch_graph":"https://pith.science/api/pith-number/INEJKZ4VHRWCNLEFY7LXKFEPFT/graph.json","fetch_events":"https://pith.science/api/pith-number/INEJKZ4VHRWCNLEFY7LXKFEPFT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/INEJKZ4VHRWCNLEFY7LXKFEPFT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/INEJKZ4VHRWCNLEFY7LXKFEPFT/action/storage_attestation","attest_author":"https://pith.science/pith/INEJKZ4VHRWCNLEFY7LXKFEPFT/action/author_attestation","sign_citation":"https://pith.science/pith/INEJKZ4VHRWCNLEFY7LXKFEPFT/action/citation_signature","submit_replication":"https://pith.science/pith/INEJKZ4VHRWCNLEFY7LXKFEPFT/action/replication_record"}},"created_at":"2026-05-18T02:44:39.923012+00:00","updated_at":"2026-05-18T02:44:39.923012+00:00"}