{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:TUHMSGL6SNOQBUYC5LETRUF52E","short_pith_number":"pith:TUHMSGL6","canonical_record":{"source":{"id":"1601.00581","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-01-04T17:44:37Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"11090051f88ac19da6ec8199b0169667b2bcdaea566ae392a0be154c068f3c68","abstract_canon_sha256":"e98a301e144392a1713f09736f61f0be9cec13d2a3f3a27f555055358124fe34"},"schema_version":"1.0"},"canonical_sha256":"9d0ec9197e935d00d302eac938d0bdd1146570e936d09f30428cfd1927fcfc97","source":{"kind":"arxiv","id":"1601.00581","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.00581","created_at":"2026-05-18T00:34:44Z"},{"alias_kind":"arxiv_version","alias_value":"1601.00581v3","created_at":"2026-05-18T00:34:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00581","created_at":"2026-05-18T00:34:44Z"},{"alias_kind":"pith_short_12","alias_value":"TUHMSGL6SNOQ","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"TUHMSGL6SNOQBUYC","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"TUHMSGL6","created_at":"2026-05-18T12:30:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:TUHMSGL6SNOQBUYC5LETRUF52E","target":"record","payload":{"canonical_record":{"source":{"id":"1601.00581","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-01-04T17:44:37Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"11090051f88ac19da6ec8199b0169667b2bcdaea566ae392a0be154c068f3c68","abstract_canon_sha256":"e98a301e144392a1713f09736f61f0be9cec13d2a3f3a27f555055358124fe34"},"schema_version":"1.0"},"canonical_sha256":"9d0ec9197e935d00d302eac938d0bdd1146570e936d09f30428cfd1927fcfc97","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:34:44.979227Z","signature_b64":"V0A6UNOk2vA+JYvE1lR/eYfnF/rKQ1z5k/AUqjDMH9MmN6SWEeUlbmvoMQl9+IkrNWvAuFFhfe+WvGxzcxIcCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d0ec9197e935d00d302eac938d0bdd1146570e936d09f30428cfd1927fcfc97","last_reissued_at":"2026-05-18T00:34:44.978564Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:34:44.978564Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.00581","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:34:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AihxES0MhNJh9JydciECIhygCqQ3lsBgjOr7GdnXBMAvce1iGWSC6Q8rETx6TGC14w+222FsdC3Qv9bLVIg6Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T01:40:39.815004Z"},"content_sha256":"c8f805a84b56ea8274cf93aed97d4278bfc8bc1f6e0986a0e48bd36eb3b4dd38","schema_version":"1.0","event_id":"sha256:c8f805a84b56ea8274cf93aed97d4278bfc8bc1f6e0986a0e48bd36eb3b4dd38"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:TUHMSGL6SNOQBUYC5LETRUF52E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Triple crystal action in Fock spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Thomas Gerber","submitted_at":"2016-01-04T17:44:37Z","abstract_excerpt":"We make explicit a triple crystal structure on higher level Fock spaces, by investigating at the combinatorial level the actions of two affine quantum groups and of a Heisenberg algebra. To this end, we first determine a new indexation of the basis elements that makes the two quantum group crystals commute. Then, we define a so-called Heisenberg crystal, commuting with the other two. This gives new information about the representation theory of cyclotomic rational Cherednik algebras, relying on some recent results of Shan and Vasserot and of Losev. In particular, we give an explicit labelling "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00581","kind":"arxiv","version":3},"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":"2026-05-18T00:34:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ht7izDWuHmXOZRsyWh3PABQOhCZWvgvkXPJkWisNmXB8nCjRDUzH43lL8uwJkdFvjJG6DWKvQJ3iemEeHz9HCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T01:40:39.815734Z"},"content_sha256":"7869703aecb1468b89dbbb7e90f84395953fef9329589d57fcc0c37a2db30ddf","schema_version":"1.0","event_id":"sha256:7869703aecb1468b89dbbb7e90f84395953fef9329589d57fcc0c37a2db30ddf"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TUHMSGL6SNOQBUYC5LETRUF52E/bundle.json","state_url":"https://pith.science/pith/TUHMSGL6SNOQBUYC5LETRUF52E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TUHMSGL6SNOQBUYC5LETRUF52E/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":"2026-06-08T01:40:39Z","links":{"resolver":"https://pith.science/pith/TUHMSGL6SNOQBUYC5LETRUF52E","bundle":"https://pith.science/pith/TUHMSGL6SNOQBUYC5LETRUF52E/bundle.json","state":"https://pith.science/pith/TUHMSGL6SNOQBUYC5LETRUF52E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TUHMSGL6SNOQBUYC5LETRUF52E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:TUHMSGL6SNOQBUYC5LETRUF52E","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":"e98a301e144392a1713f09736f61f0be9cec13d2a3f3a27f555055358124fe34","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-01-04T17:44:37Z","title_canon_sha256":"11090051f88ac19da6ec8199b0169667b2bcdaea566ae392a0be154c068f3c68"},"schema_version":"1.0","source":{"id":"1601.00581","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.00581","created_at":"2026-05-18T00:34:44Z"},{"alias_kind":"arxiv_version","alias_value":"1601.00581v3","created_at":"2026-05-18T00:34:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00581","created_at":"2026-05-18T00:34:44Z"},{"alias_kind":"pith_short_12","alias_value":"TUHMSGL6SNOQ","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"TUHMSGL6SNOQBUYC","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"TUHMSGL6","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:7869703aecb1468b89dbbb7e90f84395953fef9329589d57fcc0c37a2db30ddf","target":"graph","created_at":"2026-05-18T00:34:44Z","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"},"paper":{"abstract_excerpt":"We make explicit a triple crystal structure on higher level Fock spaces, by investigating at the combinatorial level the actions of two affine quantum groups and of a Heisenberg algebra. To this end, we first determine a new indexation of the basis elements that makes the two quantum group crystals commute. Then, we define a so-called Heisenberg crystal, commuting with the other two. This gives new information about the representation theory of cyclotomic rational Cherednik algebras, relying on some recent results of Shan and Vasserot and of Losev. In particular, we give an explicit labelling ","authors_text":"Thomas Gerber","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-01-04T17:44:37Z","title":"Triple crystal action in Fock spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00581","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:c8f805a84b56ea8274cf93aed97d4278bfc8bc1f6e0986a0e48bd36eb3b4dd38","target":"record","created_at":"2026-05-18T00:34:44Z","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":"e98a301e144392a1713f09736f61f0be9cec13d2a3f3a27f555055358124fe34","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-01-04T17:44:37Z","title_canon_sha256":"11090051f88ac19da6ec8199b0169667b2bcdaea566ae392a0be154c068f3c68"},"schema_version":"1.0","source":{"id":"1601.00581","kind":"arxiv","version":3}},"canonical_sha256":"9d0ec9197e935d00d302eac938d0bdd1146570e936d09f30428cfd1927fcfc97","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9d0ec9197e935d00d302eac938d0bdd1146570e936d09f30428cfd1927fcfc97","first_computed_at":"2026-05-18T00:34:44.978564Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:34:44.978564Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"V0A6UNOk2vA+JYvE1lR/eYfnF/rKQ1z5k/AUqjDMH9MmN6SWEeUlbmvoMQl9+IkrNWvAuFFhfe+WvGxzcxIcCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:34:44.979227Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.00581","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c8f805a84b56ea8274cf93aed97d4278bfc8bc1f6e0986a0e48bd36eb3b4dd38","sha256:7869703aecb1468b89dbbb7e90f84395953fef9329589d57fcc0c37a2db30ddf"],"state_sha256":"8c1b2ed1e2a3b2947594d60570b79d03246fd3a4201915549a58aacdf0ec33c6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0fo2G5y+6sdfa906uhIP+LcQV/4m0QFz6K+FDq8/otxRPiUa32g+Z5wDLkXE2Ds/TtL4qUkLK56jfcKd6j0CDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T01:40:39.819367Z","bundle_sha256":"157e892f75e6741bf11d51a75de43faafd5ecde0eaebe975ef10db86f05f0cfe"}}