{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:43QHEQIODMFBXKSS4MHPSJUB5V","short_pith_number":"pith:43QHEQIO","canonical_record":{"source":{"id":"0909.1126","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-09-07T01:38:08Z","cross_cats_sorted":[],"title_canon_sha256":"3d7ed553f7a6ddca991cb3db6ee2865de39add96af45fa8781d6e116fd033007","abstract_canon_sha256":"5696ab287dd2856773657371d41037c172db14786f8708e0d2ba19250730b0e7"},"schema_version":"1.0"},"canonical_sha256":"e6e072410e1b0a1baa52e30ef92681ed730c6b67075102f63491a75aa49f90aa","source":{"kind":"arxiv","id":"0909.1126","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.1126","created_at":"2026-05-18T04:31:51Z"},{"alias_kind":"arxiv_version","alias_value":"0909.1126v2","created_at":"2026-05-18T04:31:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.1126","created_at":"2026-05-18T04:31:51Z"},{"alias_kind":"pith_short_12","alias_value":"43QHEQIODMFB","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"43QHEQIODMFBXKSS","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"43QHEQIO","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:43QHEQIODMFBXKSS4MHPSJUB5V","target":"record","payload":{"canonical_record":{"source":{"id":"0909.1126","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-09-07T01:38:08Z","cross_cats_sorted":[],"title_canon_sha256":"3d7ed553f7a6ddca991cb3db6ee2865de39add96af45fa8781d6e116fd033007","abstract_canon_sha256":"5696ab287dd2856773657371d41037c172db14786f8708e0d2ba19250730b0e7"},"schema_version":"1.0"},"canonical_sha256":"e6e072410e1b0a1baa52e30ef92681ed730c6b67075102f63491a75aa49f90aa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:51.206146Z","signature_b64":"nEjSLMWFS0TM2hVK70Ryckxicl3dLgr0O1H9xhkpsDaSds2R37mhXIWlys0YuXf/CqlJJsI297OZFNdevbK8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e6e072410e1b0a1baa52e30ef92681ed730c6b67075102f63491a75aa49f90aa","last_reissued_at":"2026-05-18T04:31:51.205529Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:51.205529Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0909.1126","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:31:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ygN4qc5mpSksqwFuHEvZRXX+ZC5pNK4GHpUjzxBVykEjmq+y7toZsW//vm1HngDFJ1rIsfyCTrZwOancw9twCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T14:15:55.627522Z"},"content_sha256":"1286d07613efe14dc700eebd2b9af8d14e8c88059ee733a6c856ea56952b01b2","schema_version":"1.0","event_id":"sha256:1286d07613efe14dc700eebd2b9af8d14e8c88059ee733a6c856ea56952b01b2"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:43QHEQIODMFBXKSS4MHPSJUB5V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Crystal duality and Littlewood-Richardson rule of extremal weight crystals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Jae-Hoon Kwon","submitted_at":"2009-09-07T01:38:08Z","abstract_excerpt":"We consider a category of $\\gl_\\infty$-crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti-isomorphic to an Ore extension of the character ring of integrable lowest weight $\\gl_\\infty$-modules with respect to derivations shifting the characters of fundamental modules. A Littlewood-Richardson rule of extremal weight crystals with non-negative level is described ex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.1126","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:31:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eNeRzKfuwZuBoL6rqOQke4GA/b48KPux3yHgFNMEdccmkBYmpEJPRNZ2hcQYEKUuhnrT2S+qZLzV+A5RSW2MAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-22T14:15:55.627936Z"},"content_sha256":"b3d8606d1cd3779aed3df160783661c8bfca76eb90ceb4a9da8e06938dbc45fb","schema_version":"1.0","event_id":"sha256:b3d8606d1cd3779aed3df160783661c8bfca76eb90ceb4a9da8e06938dbc45fb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/43QHEQIODMFBXKSS4MHPSJUB5V/bundle.json","state_url":"https://pith.science/pith/43QHEQIODMFBXKSS4MHPSJUB5V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/43QHEQIODMFBXKSS4MHPSJUB5V/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-22T14:15:55Z","links":{"resolver":"https://pith.science/pith/43QHEQIODMFBXKSS4MHPSJUB5V","bundle":"https://pith.science/pith/43QHEQIODMFBXKSS4MHPSJUB5V/bundle.json","state":"https://pith.science/pith/43QHEQIODMFBXKSS4MHPSJUB5V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/43QHEQIODMFBXKSS4MHPSJUB5V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:43QHEQIODMFBXKSS4MHPSJUB5V","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":"5696ab287dd2856773657371d41037c172db14786f8708e0d2ba19250730b0e7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-09-07T01:38:08Z","title_canon_sha256":"3d7ed553f7a6ddca991cb3db6ee2865de39add96af45fa8781d6e116fd033007"},"schema_version":"1.0","source":{"id":"0909.1126","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.1126","created_at":"2026-05-18T04:31:51Z"},{"alias_kind":"arxiv_version","alias_value":"0909.1126v2","created_at":"2026-05-18T04:31:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.1126","created_at":"2026-05-18T04:31:51Z"},{"alias_kind":"pith_short_12","alias_value":"43QHEQIODMFB","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"43QHEQIODMFBXKSS","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"43QHEQIO","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:b3d8606d1cd3779aed3df160783661c8bfca76eb90ceb4a9da8e06938dbc45fb","target":"graph","created_at":"2026-05-18T04:31:51Z","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 consider a category of $\\gl_\\infty$-crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti-isomorphic to an Ore extension of the character ring of integrable lowest weight $\\gl_\\infty$-modules with respect to derivations shifting the characters of fundamental modules. A Littlewood-Richardson rule of extremal weight crystals with non-negative level is described ex","authors_text":"Jae-Hoon Kwon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-09-07T01:38:08Z","title":"Crystal duality and Littlewood-Richardson rule of extremal weight crystals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.1126","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":"sha256:1286d07613efe14dc700eebd2b9af8d14e8c88059ee733a6c856ea56952b01b2","target":"record","created_at":"2026-05-18T04:31:51Z","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":"5696ab287dd2856773657371d41037c172db14786f8708e0d2ba19250730b0e7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2009-09-07T01:38:08Z","title_canon_sha256":"3d7ed553f7a6ddca991cb3db6ee2865de39add96af45fa8781d6e116fd033007"},"schema_version":"1.0","source":{"id":"0909.1126","kind":"arxiv","version":2}},"canonical_sha256":"e6e072410e1b0a1baa52e30ef92681ed730c6b67075102f63491a75aa49f90aa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e6e072410e1b0a1baa52e30ef92681ed730c6b67075102f63491a75aa49f90aa","first_computed_at":"2026-05-18T04:31:51.205529Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:31:51.205529Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nEjSLMWFS0TM2hVK70Ryckxicl3dLgr0O1H9xhkpsDaSds2R37mhXIWlys0YuXf/CqlJJsI297OZFNdevbK8CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:31:51.206146Z","signed_message":"canonical_sha256_bytes"},"source_id":"0909.1126","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1286d07613efe14dc700eebd2b9af8d14e8c88059ee733a6c856ea56952b01b2","sha256:b3d8606d1cd3779aed3df160783661c8bfca76eb90ceb4a9da8e06938dbc45fb"],"state_sha256":"fea11db2537dc697f4043a863e231e3e0f870c7a0dc6a341b1b6d07f271cc0f6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Q3EVsM7tWhPjbv2NMkmGBUS6cfPMxYuh12+8JpilzwYBtYcvjilgPzrsYw4KRVKthb7QIbdm8rj1nSxWGfvCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-22T14:15:55.630252Z","bundle_sha256":"d3fa1e76f3efe74611ea013d1df7ec74e6f89b2ddd9aa2b5ad6c44da3b7379d8"}}