{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:BV4TZZWSB44PHN22XZXKDT4UFB","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":"668167995a1b7709cc255513880e67c4f8469a2d460a77e21fb8a4952b53d755","cross_cats_sorted":[],"license":"","primary_cat":"math.RT","submitted_at":"2007-09-20T13:12:05Z","title_canon_sha256":"6f6eabccfb6b18a514839a5cee06859b45003f7217edbe358211e1d3c9c5945b"},"schema_version":"1.0","source":{"id":"0709.3223","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0709.3223","created_at":"2026-05-17T23:47:32Z"},{"alias_kind":"arxiv_version","alias_value":"0709.3223v2","created_at":"2026-05-17T23:47:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0709.3223","created_at":"2026-05-17T23:47:32Z"},{"alias_kind":"pith_short_12","alias_value":"BV4TZZWSB44P","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"BV4TZZWSB44PHN22","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"BV4TZZWS","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:84e534447d495617f46458f4359ab33be8c3e3e7edb30a98e2683b72622b5669","target":"graph","created_at":"2026-05-17T23:47:32Z","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":"We show that for general linear groups ${\\rm GL}_n(q)$ as well as for $q$-Schur algebras the knowledge of the modular Alvis-Curtis duality over fields of characteristic $\\ell$, $\\ell \\nmid q$, is equivalent to the knowledge of the decomposition numbers.","authors_text":"Bernd Ackermann, Sibylle Schroll","cross_cats":[],"headline":"","license":"","primary_cat":"math.RT","submitted_at":"2007-09-20T13:12:05Z","title":"On decomposition numbers and Alvis-Curtis duality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.3223","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:aff5a3dfbd506af0857447fa611e659a6f4a87376780ab4abde422b3c02f774d","target":"record","created_at":"2026-05-17T23:47:32Z","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":"668167995a1b7709cc255513880e67c4f8469a2d460a77e21fb8a4952b53d755","cross_cats_sorted":[],"license":"","primary_cat":"math.RT","submitted_at":"2007-09-20T13:12:05Z","title_canon_sha256":"6f6eabccfb6b18a514839a5cee06859b45003f7217edbe358211e1d3c9c5945b"},"schema_version":"1.0","source":{"id":"0709.3223","kind":"arxiv","version":2}},"canonical_sha256":"0d793ce6d20f38f3b75abe6ea1cf94284296c85b74bd37df0327a243370619e6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d793ce6d20f38f3b75abe6ea1cf94284296c85b74bd37df0327a243370619e6","first_computed_at":"2026-05-17T23:47:32.071169Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:47:32.071169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VbZh2fi/kosrlNtNDUQOZzm2hJ0NknAJbVkHBHvTXfH/FGBo2sluzlRXCw8lE+5EE8unePs4zwNfv49vJNsUAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:47:32.071799Z","signed_message":"canonical_sha256_bytes"},"source_id":"0709.3223","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:aff5a3dfbd506af0857447fa611e659a6f4a87376780ab4abde422b3c02f774d","sha256:84e534447d495617f46458f4359ab33be8c3e3e7edb30a98e2683b72622b5669"],"state_sha256":"3360d3dca5cb0e4f025cfc9aa146c8e1883c3858fefe9f76a68d24b7a89e2206"}