{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:4TNY4SWHGHNEGHUMOZOCKFCDFH","short_pith_number":"pith:4TNY4SWH","schema_version":"1.0","canonical_sha256":"e4db8e4ac731da431e8c765c25144329c5904ae2417279dcb12d5f164490e4c6","source":{"kind":"arxiv","id":"1009.0029","version":2},"attestation_state":"computed","paper":{"title":"Idempotents in representation rings of quivers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Ralf Schiffler, Ryan Kinser","submitted_at":"2010-08-31T20:54:59Z","abstract_excerpt":"For an acyclic quiver Q, we solve the Clebsch-Gordan problem for the projective representations by computing the multiplicity of a given indecomposable projective in the tensor product of two indecomposable projectives. Motivated by this problem for arbitrary representations, we study idempotents in the representation ring of Q (the free abelian group on the indecomposable representations, with multiplication given by tensor product). We give a general technique for constructing such idempotents and for decomposing the representation ring into a direct product of ideals, utilizing morphisms be"},"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":"1009.0029","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-08-31T20:54:59Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"a7204702e14d6cdbf359d3d772c6d1863486a98ea2d851e1cfb015f48de2ec3e","abstract_canon_sha256":"7c68401fd9d4327e09304e3f6dd9f8a508e102017802f012572193c50c4521c2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:35.400265Z","signature_b64":"VPwkdU2vfrEdpVc42+osiXqpxg1gtGGtOrmmHxGqp8fKxlbAYq4O2MGI0+gshiB7AqjSaGtMVmuJ4ls6UATzAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e4db8e4ac731da431e8c765c25144329c5904ae2417279dcb12d5f164490e4c6","last_reissued_at":"2026-05-18T03:12:35.399633Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:35.399633Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Idempotents in representation rings of quivers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Ralf Schiffler, Ryan Kinser","submitted_at":"2010-08-31T20:54:59Z","abstract_excerpt":"For an acyclic quiver Q, we solve the Clebsch-Gordan problem for the projective representations by computing the multiplicity of a given indecomposable projective in the tensor product of two indecomposable projectives. Motivated by this problem for arbitrary representations, we study idempotents in the representation ring of Q (the free abelian group on the indecomposable representations, with multiplication given by tensor product). We give a general technique for constructing such idempotents and for decomposing the representation ring into a direct product of ideals, utilizing morphisms be"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0029","kind":"arxiv","version":2},"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":"1009.0029","created_at":"2026-05-18T03:12:35.399743+00:00"},{"alias_kind":"arxiv_version","alias_value":"1009.0029v2","created_at":"2026-05-18T03:12:35.399743+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.0029","created_at":"2026-05-18T03:12:35.399743+00:00"},{"alias_kind":"pith_short_12","alias_value":"4TNY4SWHGHNE","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_16","alias_value":"4TNY4SWHGHNEGHUM","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_8","alias_value":"4TNY4SWH","created_at":"2026-05-18T12:26:04.259169+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/4TNY4SWHGHNEGHUMOZOCKFCDFH","json":"https://pith.science/pith/4TNY4SWHGHNEGHUMOZOCKFCDFH.json","graph_json":"https://pith.science/api/pith-number/4TNY4SWHGHNEGHUMOZOCKFCDFH/graph.json","events_json":"https://pith.science/api/pith-number/4TNY4SWHGHNEGHUMOZOCKFCDFH/events.json","paper":"https://pith.science/paper/4TNY4SWH"},"agent_actions":{"view_html":"https://pith.science/pith/4TNY4SWHGHNEGHUMOZOCKFCDFH","download_json":"https://pith.science/pith/4TNY4SWHGHNEGHUMOZOCKFCDFH.json","view_paper":"https://pith.science/paper/4TNY4SWH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1009.0029&json=true","fetch_graph":"https://pith.science/api/pith-number/4TNY4SWHGHNEGHUMOZOCKFCDFH/graph.json","fetch_events":"https://pith.science/api/pith-number/4TNY4SWHGHNEGHUMOZOCKFCDFH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4TNY4SWHGHNEGHUMOZOCKFCDFH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4TNY4SWHGHNEGHUMOZOCKFCDFH/action/storage_attestation","attest_author":"https://pith.science/pith/4TNY4SWHGHNEGHUMOZOCKFCDFH/action/author_attestation","sign_citation":"https://pith.science/pith/4TNY4SWHGHNEGHUMOZOCKFCDFH/action/citation_signature","submit_replication":"https://pith.science/pith/4TNY4SWHGHNEGHUMOZOCKFCDFH/action/replication_record"}},"created_at":"2026-05-18T03:12:35.399743+00:00","updated_at":"2026-05-18T03:12:35.399743+00:00"}