{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:BRLFFWJYZULRMGRRMK5MGDBZOV","short_pith_number":"pith:BRLFFWJY","canonical_record":{"source":{"id":"1707.02565","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-07-09T11:49:15Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a1fdbd3baa563531b08cbc989f08a6b7e69cae7576c50987184638f814ac5a8b","abstract_canon_sha256":"2de604175de0a4a87c49697344848032ef8974ee3fe4f7a41e7bb8b0a07fd630"},"schema_version":"1.0"},"canonical_sha256":"0c5652d938cd17161a3162bac30c397544db1a591438652a0fd237b079befde0","source":{"kind":"arxiv","id":"1707.02565","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.02565","created_at":"2026-05-18T00:17:13Z"},{"alias_kind":"arxiv_version","alias_value":"1707.02565v2","created_at":"2026-05-18T00:17:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.02565","created_at":"2026-05-18T00:17:13Z"},{"alias_kind":"pith_short_12","alias_value":"BRLFFWJYZULR","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BRLFFWJYZULRMGRR","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BRLFFWJY","created_at":"2026-05-18T12:31:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:BRLFFWJYZULRMGRRMK5MGDBZOV","target":"record","payload":{"canonical_record":{"source":{"id":"1707.02565","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-07-09T11:49:15Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a1fdbd3baa563531b08cbc989f08a6b7e69cae7576c50987184638f814ac5a8b","abstract_canon_sha256":"2de604175de0a4a87c49697344848032ef8974ee3fe4f7a41e7bb8b0a07fd630"},"schema_version":"1.0"},"canonical_sha256":"0c5652d938cd17161a3162bac30c397544db1a591438652a0fd237b079befde0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:13.406041Z","signature_b64":"l4riaOo9eZN1/aV5vaf10Ee33Ycf7Gs7SncfAcnpNzyvtKtRfWBNqNNDeF0cilexVWwl7d09tNXXWsfmoix2Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c5652d938cd17161a3162bac30c397544db1a591438652a0fd237b079befde0","last_reissued_at":"2026-05-18T00:17:13.405350Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:13.405350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1707.02565","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":"2026-05-18T00:17:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FCdWuHsR2WrClkOirSeVWgJfsOHx/zChG+HKk2YgyNUJWvPM0SP1/8AhSUkV2MHxHzU2RLy8J5pHKnSo5km4Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T09:08:56.733729Z"},"content_sha256":"5be0ae3cbd7f242e5c3c284f60f27214877cd7a8c786acc92dfdac3400e9e159","schema_version":"1.0","event_id":"sha256:5be0ae3cbd7f242e5c3c284f60f27214877cd7a8c786acc92dfdac3400e9e159"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:BRLFFWJYZULRMGRRMK5MGDBZOV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Gelfand-Kirillov Dimensions of Highest Weight Harish-Chandra Modules for $SU(p,q)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Xun Xie, Zhanqiang Bai","submitted_at":"2017-07-09T11:49:15Z","abstract_excerpt":"Let $ (G,K) $ be an irreducible Hermitian symmetric pair of non-compact type with $G=SU(p,q)$, and let $ \\lambda $ be an integral weight such that the simple highest weight module $ L(\\lambda) $ is a Harish-Chandra $ (\\mathfrak{g},K) $-module. We give a combinatoric algorithm for the Gelfand-Kirillov dimension of $ L(\\lambda) $. This enables us to prove that the Gelfand-Kirillov dimension of $ L(\\lambda) $ decreases as the integer $ \\langle\\lambda+\\rho,\\beta^\\vee\\rangle $ increases, where $\\rho$ is the half sum of positive roots and $\\beta$ is the maximal noncompact root. As a byproduct, we ob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.02565","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":"2026-05-18T00:17:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"epYUnynE67WuV2P3OFE7GUHgL3BNC42qLypcwEA2Rz5LqA4sxixm5Jz7OJ9ZuC0XoVKsG5lXXk0aO1edTa1xAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T09:08:56.734440Z"},"content_sha256":"5e3cb81f291b4b7233c91096580986d2bac7f6e5cbac2aeb723d36d34f0a4cf0","schema_version":"1.0","event_id":"sha256:5e3cb81f291b4b7233c91096580986d2bac7f6e5cbac2aeb723d36d34f0a4cf0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BRLFFWJYZULRMGRRMK5MGDBZOV/bundle.json","state_url":"https://pith.science/pith/BRLFFWJYZULRMGRRMK5MGDBZOV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BRLFFWJYZULRMGRRMK5MGDBZOV/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":"2026-05-26T09:08:56Z","links":{"resolver":"https://pith.science/pith/BRLFFWJYZULRMGRRMK5MGDBZOV","bundle":"https://pith.science/pith/BRLFFWJYZULRMGRRMK5MGDBZOV/bundle.json","state":"https://pith.science/pith/BRLFFWJYZULRMGRRMK5MGDBZOV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BRLFFWJYZULRMGRRMK5MGDBZOV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:BRLFFWJYZULRMGRRMK5MGDBZOV","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":"2de604175de0a4a87c49697344848032ef8974ee3fe4f7a41e7bb8b0a07fd630","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-07-09T11:49:15Z","title_canon_sha256":"a1fdbd3baa563531b08cbc989f08a6b7e69cae7576c50987184638f814ac5a8b"},"schema_version":"1.0","source":{"id":"1707.02565","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.02565","created_at":"2026-05-18T00:17:13Z"},{"alias_kind":"arxiv_version","alias_value":"1707.02565v2","created_at":"2026-05-18T00:17:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.02565","created_at":"2026-05-18T00:17:13Z"},{"alias_kind":"pith_short_12","alias_value":"BRLFFWJYZULR","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_16","alias_value":"BRLFFWJYZULRMGRR","created_at":"2026-05-18T12:31:08Z"},{"alias_kind":"pith_short_8","alias_value":"BRLFFWJY","created_at":"2026-05-18T12:31:08Z"}],"graph_snapshots":[{"event_id":"sha256:5e3cb81f291b4b7233c91096580986d2bac7f6e5cbac2aeb723d36d34f0a4cf0","target":"graph","created_at":"2026-05-18T00:17:13Z","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":"Let $ (G,K) $ be an irreducible Hermitian symmetric pair of non-compact type with $G=SU(p,q)$, and let $ \\lambda $ be an integral weight such that the simple highest weight module $ L(\\lambda) $ is a Harish-Chandra $ (\\mathfrak{g},K) $-module. We give a combinatoric algorithm for the Gelfand-Kirillov dimension of $ L(\\lambda) $. This enables us to prove that the Gelfand-Kirillov dimension of $ L(\\lambda) $ decreases as the integer $ \\langle\\lambda+\\rho,\\beta^\\vee\\rangle $ increases, where $\\rho$ is the half sum of positive roots and $\\beta$ is the maximal noncompact root. As a byproduct, we ob","authors_text":"Xun Xie, Zhanqiang Bai","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-07-09T11:49:15Z","title":"Gelfand-Kirillov Dimensions of Highest Weight Harish-Chandra Modules for $SU(p,q)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.02565","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":"sha256:5be0ae3cbd7f242e5c3c284f60f27214877cd7a8c786acc92dfdac3400e9e159","target":"record","created_at":"2026-05-18T00:17:13Z","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":"2de604175de0a4a87c49697344848032ef8974ee3fe4f7a41e7bb8b0a07fd630","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-07-09T11:49:15Z","title_canon_sha256":"a1fdbd3baa563531b08cbc989f08a6b7e69cae7576c50987184638f814ac5a8b"},"schema_version":"1.0","source":{"id":"1707.02565","kind":"arxiv","version":2}},"canonical_sha256":"0c5652d938cd17161a3162bac30c397544db1a591438652a0fd237b079befde0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0c5652d938cd17161a3162bac30c397544db1a591438652a0fd237b079befde0","first_computed_at":"2026-05-18T00:17:13.405350Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:13.405350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"l4riaOo9eZN1/aV5vaf10Ee33Ycf7Gs7SncfAcnpNzyvtKtRfWBNqNNDeF0cilexVWwl7d09tNXXWsfmoix2Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:13.406041Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.02565","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5be0ae3cbd7f242e5c3c284f60f27214877cd7a8c786acc92dfdac3400e9e159","sha256:5e3cb81f291b4b7233c91096580986d2bac7f6e5cbac2aeb723d36d34f0a4cf0"],"state_sha256":"169f227334d525bbff8039593062aeb7f636c99d42e343fd58fb85eaffe13eb1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4hBZUpYb2KbnR8iRy6TQ5F04cWph5EWVctaqbq121mSpMM8Vj9ZsuHTedy7CEBedhVdytuDwKQX9fVfvthncAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T09:08:56.737708Z","bundle_sha256":"02fa84ea45d9675d0dcb5f8496ca1fd9d1a4e31e8bad595c44ed10076af8d6f8"}}