{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:4TNY4SWHGHNEGHUMOZOCKFCDFH","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7c68401fd9d4327e09304e3f6dd9f8a508e102017802f012572193c50c4521c2","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-08-31T20:54:59Z","title_canon_sha256":"a7204702e14d6cdbf359d3d772c6d1863486a98ea2d851e1cfb015f48de2ec3e"},"schema_version":"1.0","source":{"id":"1009.0029","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1009.0029","created_at":"2026-05-18T03:12:35Z"},{"alias_kind":"arxiv_version","alias_value":"1009.0029v2","created_at":"2026-05-18T03:12:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1009.0029","created_at":"2026-05-18T03:12:35Z"},{"alias_kind":"pith_short_12","alias_value":"4TNY4SWHGHNE","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"4TNY4SWHGHNEGHUM","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"4TNY4SWH","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:972e5060ed9343ebe35101510a3678122ebdeff4387030f78895198aca683dc6","target":"graph","created_at":"2026-05-18T03:12:35Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Ralf Schiffler, Ryan Kinser","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-08-31T20:54:59Z","title":"Idempotents in representation rings of quivers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.0029","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:64df3090ad04cbd4556b60d3d42ac19f2d7f28bc3a6d6c13f1fc284360be6873","target":"record","created_at":"2026-05-18T03:12:35Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7c68401fd9d4327e09304e3f6dd9f8a508e102017802f012572193c50c4521c2","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-08-31T20:54:59Z","title_canon_sha256":"a7204702e14d6cdbf359d3d772c6d1863486a98ea2d851e1cfb015f48de2ec3e"},"schema_version":"1.0","source":{"id":"1009.0029","kind":"arxiv","version":2}},"canonical_sha256":"e4db8e4ac731da431e8c765c25144329c5904ae2417279dcb12d5f164490e4c6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e4db8e4ac731da431e8c765c25144329c5904ae2417279dcb12d5f164490e4c6","first_computed_at":"2026-05-18T03:12:35.399633Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:12:35.399633Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VPwkdU2vfrEdpVc42+osiXqpxg1gtGGtOrmmHxGqp8fKxlbAYq4O2MGI0+gshiB7AqjSaGtMVmuJ4ls6UATzAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:12:35.400265Z","signed_message":"canonical_sha256_bytes"},"source_id":"1009.0029","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:64df3090ad04cbd4556b60d3d42ac19f2d7f28bc3a6d6c13f1fc284360be6873","sha256:972e5060ed9343ebe35101510a3678122ebdeff4387030f78895198aca683dc6"],"state_sha256":"603ebf156406efc70279b2cd008c068c5a7fda7486d63c34d0e1f09875714f63"}