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The exceptional representations of $Q$, that is, the indecomposable objects of ${\\rm rep}(Q)$ without self-extensions, correspond to the so-called real Schur roots of the usual root system attached to $Q$. These roots are special elements of the Grothendieck group $\\mathbb{Z}^n$ of ${\\rm rep}(Q)$. When we identify the dimension vectors of the representations (that is, the non-negative vectors of $\\mathbb{Z}^n$) up t"},"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":"1503.02054","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-03-06T20:20:38Z","cross_cats_sorted":[],"title_canon_sha256":"9834f76aa5994f1f679a8b52721d47ae69e2eb896a0f0ff5ad7d8e00082200a5","abstract_canon_sha256":"3f0f105218f69c011c07d85e0e1a4fdf0791530236d8e38f2cffd965a0f6f0f5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:26.947396Z","signature_b64":"yZ6LqhNsZiChtFD2TRuwuXrQvmYAPy0J9XcsmliVQ5u0KxkprV/bTk23vsi6/vIiq1UZQ1hT0/+MF9WSKPGIDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc1cdf2abb2b1752d6e708f801344a523ad4b3d8fb5517083f29d3e473f2d05d","last_reissued_at":"2026-05-18T02:25:26.946822Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:26.946822Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Accumulation points of real Schur roots","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Charles Paquette","submitted_at":"2015-03-06T20:20:38Z","abstract_excerpt":"Let $k$ be an algebraically closed field and $Q$ be an acyclic quiver with $n$ vertices. Consider the category ${\\rm rep}(Q)$ of finite dimensional representations of $Q$ over $k$. The exceptional representations of $Q$, that is, the indecomposable objects of ${\\rm rep}(Q)$ without self-extensions, correspond to the so-called real Schur roots of the usual root system attached to $Q$. These roots are special elements of the Grothendieck group $\\mathbb{Z}^n$ of ${\\rm rep}(Q)$. When we identify the dimension vectors of the representations (that is, the non-negative vectors of $\\mathbb{Z}^n$) up t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.02054","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":"1503.02054","created_at":"2026-05-18T02:25:26.946904+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.02054v1","created_at":"2026-05-18T02:25:26.946904+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.02054","created_at":"2026-05-18T02:25:26.946904+00:00"},{"alias_kind":"pith_short_12","alias_value":"7QON6KV3FMLV","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"7QON6KV3FMLVFVXH","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"7QON6KV3","created_at":"2026-05-18T12:29:10.953037+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/7QON6KV3FMLVFVXHBD4ACNCKKI","json":"https://pith.science/pith/7QON6KV3FMLVFVXHBD4ACNCKKI.json","graph_json":"https://pith.science/api/pith-number/7QON6KV3FMLVFVXHBD4ACNCKKI/graph.json","events_json":"https://pith.science/api/pith-number/7QON6KV3FMLVFVXHBD4ACNCKKI/events.json","paper":"https://pith.science/paper/7QON6KV3"},"agent_actions":{"view_html":"https://pith.science/pith/7QON6KV3FMLVFVXHBD4ACNCKKI","download_json":"https://pith.science/pith/7QON6KV3FMLVFVXHBD4ACNCKKI.json","view_paper":"https://pith.science/paper/7QON6KV3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.02054&json=true","fetch_graph":"https://pith.science/api/pith-number/7QON6KV3FMLVFVXHBD4ACNCKKI/graph.json","fetch_events":"https://pith.science/api/pith-number/7QON6KV3FMLVFVXHBD4ACNCKKI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7QON6KV3FMLVFVXHBD4ACNCKKI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7QON6KV3FMLVFVXHBD4ACNCKKI/action/storage_attestation","attest_author":"https://pith.science/pith/7QON6KV3FMLVFVXHBD4ACNCKKI/action/author_attestation","sign_citation":"https://pith.science/pith/7QON6KV3FMLVFVXHBD4ACNCKKI/action/citation_signature","submit_replication":"https://pith.science/pith/7QON6KV3FMLVFVXHBD4ACNCKKI/action/replication_record"}},"created_at":"2026-05-18T02:25:26.946904+00:00","updated_at":"2026-05-18T02:25:26.946904+00:00"}