{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2004:XV2KE5LOACF5VG722DAONNVCUT","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":"d38c8ae3d7b6ade9f61726b48cfcc49ecbe87b876dd8091fb22fe5ca354a5ccf","cross_cats_sorted":["math.CO","math.CT"],"license":"","primary_cat":"math.QA","submitted_at":"2004-06-23T15:54:12Z","title_canon_sha256":"1d5c23d5042c70a3f67b1b55c9f077923d4afa4e10c377dbac9ddc1e436bf476"},"schema_version":"1.0","source":{"id":"math/0406478","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0406478","created_at":"2026-07-04T14:38:44Z"},{"alias_kind":"arxiv_version","alias_value":"math/0406478v3","created_at":"2026-07-04T14:38:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0406478","created_at":"2026-07-04T14:38:44Z"},{"alias_kind":"pith_short_12","alias_value":"XV2KE5LOACF5","created_at":"2026-07-04T14:38:44Z"},{"alias_kind":"pith_short_16","alias_value":"XV2KE5LOACF5VG72","created_at":"2026-07-04T14:38:44Z"},{"alias_kind":"pith_short_8","alias_value":"XV2KE5LO","created_at":"2026-07-04T14:38:44Z"}],"graph_snapshots":[{"event_id":"sha256:e4ec13669f7ff4e612b0ca8ab82e43168f70c44036d1761d2e172610affbdc79","target":"graph","created_at":"2026-07-04T14:38:44Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0406478/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Following an idea of A. Berenstein, we define a commutor for the category of crystals of a finite dimensional complex reductive Lie algebra. We show that this endows the category of crystals with the structure of a coboundary category. Similar to the case of braided categories, there is a group naturally acting on multiple tensor products in coboundary categories. We call this group the cactus group and identify it as the fundamental group of the moduli space of marked real genus zero stable curves.","authors_text":"Andre Henriques, Joel Kamnitzer","cross_cats":["math.CO","math.CT"],"headline":"","license":"","primary_cat":"math.QA","submitted_at":"2004-06-23T15:54:12Z","title":"Crystals and coboundary categories"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0406478","kind":"arxiv","version":3},"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":"sha256:3bef96f5226bf6e8735d31d197616e7afa39925b2b1776ff72355893a40fcd10","target":"record","created_at":"2026-07-04T14:38:44Z","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":"d38c8ae3d7b6ade9f61726b48cfcc49ecbe87b876dd8091fb22fe5ca354a5ccf","cross_cats_sorted":["math.CO","math.CT"],"license":"","primary_cat":"math.QA","submitted_at":"2004-06-23T15:54:12Z","title_canon_sha256":"1d5c23d5042c70a3f67b1b55c9f077923d4afa4e10c377dbac9ddc1e436bf476"},"schema_version":"1.0","source":{"id":"math/0406478","kind":"arxiv","version":3}},"canonical_sha256":"bd74a2756e008bda9bfad0c0e6b6a2a4cb3f2971133f60dda0830a5480ef61ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bd74a2756e008bda9bfad0c0e6b6a2a4cb3f2971133f60dda0830a5480ef61ff","first_computed_at":"2026-07-04T14:38:44.575823Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:38:44.575823Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"J2ai+LOd5BN61xh/FQtU6/VX6IPotirCbfOT+pVa3WY+B6X6Y5PtCQueyijnQRTsWv5350dY6WDpuadoUWaHBw==","signature_status":"signed_v1","signed_at":"2026-07-04T14:38:44.576266Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0406478","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3bef96f5226bf6e8735d31d197616e7afa39925b2b1776ff72355893a40fcd10","sha256:e4ec13669f7ff4e612b0ca8ab82e43168f70c44036d1761d2e172610affbdc79"],"state_sha256":"329ff9333a52b3a3a18200e8edf6717e2cfa82412b59a53dba940c52a768c95c"}