{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:KS4XVHEKEIMMJ6KFPMZPU4DPZU","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":"94e31f3976bf608bbe774d9d88fbff6474daa620788a8f9df84cd1f509d55540","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-02-13T17:15:21Z","title_canon_sha256":"e2b96381605eef634df7e1b8a23520b18320c0afb027e1af774847789916706b"},"schema_version":"1.0","source":{"id":"0902.2349","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0902.2349","created_at":"2026-05-18T03:37:06Z"},{"alias_kind":"arxiv_version","alias_value":"0902.2349v1","created_at":"2026-05-18T03:37:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0902.2349","created_at":"2026-05-18T03:37:06Z"},{"alias_kind":"pith_short_12","alias_value":"KS4XVHEKEIMM","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_16","alias_value":"KS4XVHEKEIMMJ6KF","created_at":"2026-05-18T12:26:00Z"},{"alias_kind":"pith_short_8","alias_value":"KS4XVHEK","created_at":"2026-05-18T12:26:00Z"}],"graph_snapshots":[{"event_id":"sha256:016280abca68cea193a31789aedd4299e9a1f4340ac4a799b0be7a0a350023cb","target":"graph","created_at":"2026-05-18T03:37:06Z","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":"Following the work of B. Kuelshammer, J. B. Olsson and G. R. Robinson on generalized blocks of the symmetric groups, we give a definition for the \\ell-defect of characters of the symmetric group S_n, where \\ell > 1 is an arbitrary integer. We prove that the \\ell -defect is given by an analogue of the hook-length formula, and use it to prove, when n < \\ell^2, an \\ell-version of the McKay Conjecture in S_n .","authors_text":"Jean-Baptiste Gramain","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-02-13T17:15:21Z","title":"Defect of characters of the symmetric group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0902.2349","kind":"arxiv","version":1},"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":"sha256:8f71b311f0709d1188bdf978cc8db246edb54bb33560d872c00de7fb65e0e7fb","target":"record","created_at":"2026-05-18T03:37:06Z","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":"94e31f3976bf608bbe774d9d88fbff6474daa620788a8f9df84cd1f509d55540","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2009-02-13T17:15:21Z","title_canon_sha256":"e2b96381605eef634df7e1b8a23520b18320c0afb027e1af774847789916706b"},"schema_version":"1.0","source":{"id":"0902.2349","kind":"arxiv","version":1}},"canonical_sha256":"54b97a9c8a2218c4f9457b32fa706fcd1f128c9389c0e7e5ef55d4fd2f8e3f46","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"54b97a9c8a2218c4f9457b32fa706fcd1f128c9389c0e7e5ef55d4fd2f8e3f46","first_computed_at":"2026-05-18T03:37:06.004090Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:37:06.004090Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m1lbCfejAFh2FazhaeTC/lQc3v7rcn/WHykECMVEsANRLZEHRAv0iV+LiaTr/rrJcfupsMJMnXMPrT/yOGl9BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:37:06.004864Z","signed_message":"canonical_sha256_bytes"},"source_id":"0902.2349","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8f71b311f0709d1188bdf978cc8db246edb54bb33560d872c00de7fb65e0e7fb","sha256:016280abca68cea193a31789aedd4299e9a1f4340ac4a799b0be7a0a350023cb"],"state_sha256":"4fe265b8eef64c411daaa1503cb1e5c420b550c89d057613c0267d89d8353eb4"}