{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:JMBSR5DHCNWMBIRC6PY677SK7M","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":"65d4d0a7b781a7686c2cd8800f34f7c54eaeecd1aaf91cc208fe758d1ba4864d","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-05-20T12:59:29Z","title_canon_sha256":"bc0cf68fcf5674c93a11eb52bc64a69acf52038946c44b3e391c728fb9f1e683"},"schema_version":"1.0","source":{"id":"1505.05353","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.05353","created_at":"2026-05-18T01:31:13Z"},{"alias_kind":"arxiv_version","alias_value":"1505.05353v2","created_at":"2026-05-18T01:31:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.05353","created_at":"2026-05-18T01:31:13Z"},{"alias_kind":"pith_short_12","alias_value":"JMBSR5DHCNWM","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"JMBSR5DHCNWMBIRC","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"JMBSR5DH","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:782719ab5e9464493f06cf4c7905770b47c68ccbb2554bb9305ba936a75c7756","target":"graph","created_at":"2026-05-18T01:31:13Z","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 investigate the relation between the Garside normal form for positive braids and the $2$-braid group defined by Rouquier. Inspired by work of Brav and Thomas we show that the Garside normal form is encoded in the action of the $2$-braid group on a certain categorified left cell module. This allows us to deduce the faithfulness of the $2$-braid group in finite type. We also give a new proof of Paris' theorem that the canonical map from the generalized braid monoid to its braid group is injective in arbitrary type.","authors_text":"Lars Thorge Jensen","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-05-20T12:59:29Z","title":"The 2-braid group and Garside normal form"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.05353","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":"sha256:23bbe0b7b98ec38c5ef619d0cdebb270487184427f8be861f61067bec0aa896d","target":"record","created_at":"2026-05-18T01:31:13Z","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":"65d4d0a7b781a7686c2cd8800f34f7c54eaeecd1aaf91cc208fe758d1ba4864d","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-05-20T12:59:29Z","title_canon_sha256":"bc0cf68fcf5674c93a11eb52bc64a69acf52038946c44b3e391c728fb9f1e683"},"schema_version":"1.0","source":{"id":"1505.05353","kind":"arxiv","version":2}},"canonical_sha256":"4b0328f467136cc0a222f3f1effe4afb2d80168c12ee94d48d80b4fdecc72ce9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b0328f467136cc0a222f3f1effe4afb2d80168c12ee94d48d80b4fdecc72ce9","first_computed_at":"2026-05-18T01:31:13.675864Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:13.675864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"59MLiHNsJswQkKo5atXVFSBx7AJoDEnX9v2ig/Wha4iXqsnzqzR7tqYuTAyAF67ilIxdqQgc28nWMSJlXqpAAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:13.676627Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.05353","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:23bbe0b7b98ec38c5ef619d0cdebb270487184427f8be861f61067bec0aa896d","sha256:782719ab5e9464493f06cf4c7905770b47c68ccbb2554bb9305ba936a75c7756"],"state_sha256":"6d0f50677c45761b6e3e29ea2d20c19ab562ea7179c602cb40f6a05338ccca2b"}