{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:BF4EW2MUA2JMIWRCKERM3TJTB4","short_pith_number":"pith:BF4EW2MU","schema_version":"1.0","canonical_sha256":"09784b69940692c45a225122cdcd330f2615e319367007baf13a5fe0a36a6b15","source":{"kind":"arxiv","id":"math/0608288","version":2},"attestation_state":"computed","paper":{"title":"The Combinatorics of Quiver Representations","license":"","headline":"","cross_cats":["math.AC","math.AG","math.CO"],"primary_cat":"math.RT","authors_text":"Harm Derksen, Jerzy Weyman","submitted_at":"2006-08-11T17:44:55Z","abstract_excerpt":"We give a description of faces of all codimensions for the cones of weights of rings of semi-invariants of quivers. For a triple flag quiver and faces of codimension 1 this reduces to the result of Knutson-Tao-Woodward on the facets of the Klyachko cone. We give new applications to Littlewood-Richardson coefficients, including a product formula for LR-coefficients corresponding to triples of partitions lying on a wall of the Klyachko cone. We systematically review and develop the necessary methods (exceptional and Schur sequences, orthogonal categories, semi-stable decompositions, GIT quotient"},"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":"math/0608288","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.RT","submitted_at":"2006-08-11T17:44:55Z","cross_cats_sorted":["math.AC","math.AG","math.CO"],"title_canon_sha256":"83b1bf6ac5451a1b2f3f1fb1d4af8782f781daa25680b62eb920924c6869e436","abstract_canon_sha256":"52995f5a2ea900ac208c5ccb40c1882b383f8f5330bd621876e7ab036fa80189"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:08:53.323544Z","signature_b64":"7VNEs/yq/Tf32RkPQ9hNPGO+Qv9VNlCA4NUranEfbQM28xHVr2A+RrmcCXP3qeClUfEf7VXQGk75oxjVcvS0DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"09784b69940692c45a225122cdcd330f2615e319367007baf13a5fe0a36a6b15","last_reissued_at":"2026-05-18T04:08:53.322896Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:08:53.322896Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Combinatorics of Quiver Representations","license":"","headline":"","cross_cats":["math.AC","math.AG","math.CO"],"primary_cat":"math.RT","authors_text":"Harm Derksen, Jerzy Weyman","submitted_at":"2006-08-11T17:44:55Z","abstract_excerpt":"We give a description of faces of all codimensions for the cones of weights of rings of semi-invariants of quivers. For a triple flag quiver and faces of codimension 1 this reduces to the result of Knutson-Tao-Woodward on the facets of the Klyachko cone. We give new applications to Littlewood-Richardson coefficients, including a product formula for LR-coefficients corresponding to triples of partitions lying on a wall of the Klyachko cone. We systematically review and develop the necessary methods (exceptional and Schur sequences, orthogonal categories, semi-stable decompositions, GIT quotient"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0608288","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":"math/0608288","created_at":"2026-05-18T04:08:53.322992+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0608288v2","created_at":"2026-05-18T04:08:53.322992+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0608288","created_at":"2026-05-18T04:08:53.322992+00:00"},{"alias_kind":"pith_short_12","alias_value":"BF4EW2MUA2JM","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"BF4EW2MUA2JMIWRC","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"BF4EW2MU","created_at":"2026-05-18T12:25:53.939244+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/BF4EW2MUA2JMIWRCKERM3TJTB4","json":"https://pith.science/pith/BF4EW2MUA2JMIWRCKERM3TJTB4.json","graph_json":"https://pith.science/api/pith-number/BF4EW2MUA2JMIWRCKERM3TJTB4/graph.json","events_json":"https://pith.science/api/pith-number/BF4EW2MUA2JMIWRCKERM3TJTB4/events.json","paper":"https://pith.science/paper/BF4EW2MU"},"agent_actions":{"view_html":"https://pith.science/pith/BF4EW2MUA2JMIWRCKERM3TJTB4","download_json":"https://pith.science/pith/BF4EW2MUA2JMIWRCKERM3TJTB4.json","view_paper":"https://pith.science/paper/BF4EW2MU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0608288&json=true","fetch_graph":"https://pith.science/api/pith-number/BF4EW2MUA2JMIWRCKERM3TJTB4/graph.json","fetch_events":"https://pith.science/api/pith-number/BF4EW2MUA2JMIWRCKERM3TJTB4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BF4EW2MUA2JMIWRCKERM3TJTB4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BF4EW2MUA2JMIWRCKERM3TJTB4/action/storage_attestation","attest_author":"https://pith.science/pith/BF4EW2MUA2JMIWRCKERM3TJTB4/action/author_attestation","sign_citation":"https://pith.science/pith/BF4EW2MUA2JMIWRCKERM3TJTB4/action/citation_signature","submit_replication":"https://pith.science/pith/BF4EW2MUA2JMIWRCKERM3TJTB4/action/replication_record"}},"created_at":"2026-05-18T04:08:53.322992+00:00","updated_at":"2026-05-18T04:08:53.322992+00:00"}