{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:L7KF7M5MQR4VEYT7KUY2CFRFAC","short_pith_number":"pith:L7KF7M5M","canonical_record":{"source":{"id":"1708.04428","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-08-15T08:23:26Z","cross_cats_sorted":[],"title_canon_sha256":"9a53fda2c87c56b7666a8b737900e280b76387309804f4ed1554febaf07d770f","abstract_canon_sha256":"183e38dc9d2a9bd525c4e0385d1c8c132340b4a79425fca90d15ee3effeee84c"},"schema_version":"1.0"},"canonical_sha256":"5fd45fb3ac847952627f5531a1162500ba81cd188e6883a51c5dac3019b70a73","source":{"kind":"arxiv","id":"1708.04428","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.04428","created_at":"2026-05-18T00:23:23Z"},{"alias_kind":"arxiv_version","alias_value":"1708.04428v2","created_at":"2026-05-18T00:23:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.04428","created_at":"2026-05-18T00:23:23Z"},{"alias_kind":"pith_short_12","alias_value":"L7KF7M5MQR4V","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_16","alias_value":"L7KF7M5MQR4VEYT7","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_8","alias_value":"L7KF7M5M","created_at":"2026-05-18T12:31:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:L7KF7M5MQR4VEYT7KUY2CFRFAC","target":"record","payload":{"canonical_record":{"source":{"id":"1708.04428","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-08-15T08:23:26Z","cross_cats_sorted":[],"title_canon_sha256":"9a53fda2c87c56b7666a8b737900e280b76387309804f4ed1554febaf07d770f","abstract_canon_sha256":"183e38dc9d2a9bd525c4e0385d1c8c132340b4a79425fca90d15ee3effeee84c"},"schema_version":"1.0"},"canonical_sha256":"5fd45fb3ac847952627f5531a1162500ba81cd188e6883a51c5dac3019b70a73","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:23.156958Z","signature_b64":"xlDM5oFVaALh+eREcjr5iWMlZpC1vLbBNY0pLsyhC4WSEO65g6pgf/bj+pHBYsaSwPu9m4VSS2FDWmvc0vFCDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5fd45fb3ac847952627f5531a1162500ba81cd188e6883a51c5dac3019b70a73","last_reissued_at":"2026-05-18T00:23:23.156248Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:23.156248Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1708.04428","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":"2026-05-18T00:23:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gay73CZiFcQggAMqJU/maoIHHVQFTmuSml7b7GJDDa4D6Sov9H5gFzWxrkcvfpZHQTStEt5IJP9Agl5dx/OlBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T17:03:08.314507Z"},"content_sha256":"de5e4165a19c317d96f0319695d585c81bc5a588dd308ba8dd93e405cad5be44","schema_version":"1.0","event_id":"sha256:de5e4165a19c317d96f0319695d585c81bc5a588dd308ba8dd93e405cad5be44"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:L7KF7M5MQR4VEYT7KUY2CFRFAC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Monoidal categories associated with strata of flag manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Euiyong Park, Masaki Kashiwara, MyungHo Kim, Se-Jin Oh","submitted_at":"2017-08-15T08:23:26Z","abstract_excerpt":"We construct a monoidal category $\\mathscr{C}_{w,v}$ which categorifies the doubly-invariant algebra $^{N'(w)}\\mathbb{C}[N]^{N(v)}$ associated with Weyl group elements $w$ and $v$. It gives, after a localization, the coordinate algebra $\\mathbb{C}[\\mathcal{R}_{w,v}]$ of the open Richardson variety associated with $w$ and $v$. The category $\\mathscr{C}_{w,v}$ is realized as a subcategory of the graded module category of a quiver Hecke algebra $R$. When $v= \\mathrm{id}$, $\\mathscr{C}_{w,v}$ is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.04428","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":"2026-05-18T00:23:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TszwkKUVor60RNn0JcxGAR6BMrXzEQvigL9ZfNmuGOGfa0nC5+KBYPHeyr2QUkbeZNCTnB25Yrjuy+BpNJSVCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T17:03:08.315152Z"},"content_sha256":"7c9a56af633dd206438e61cd2b4f52f06d14b32239bcd4e02e89bccf782c0898","schema_version":"1.0","event_id":"sha256:7c9a56af633dd206438e61cd2b4f52f06d14b32239bcd4e02e89bccf782c0898"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L7KF7M5MQR4VEYT7KUY2CFRFAC/bundle.json","state_url":"https://pith.science/pith/L7KF7M5MQR4VEYT7KUY2CFRFAC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L7KF7M5MQR4VEYT7KUY2CFRFAC/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":"2026-05-25T17:03:08Z","links":{"resolver":"https://pith.science/pith/L7KF7M5MQR4VEYT7KUY2CFRFAC","bundle":"https://pith.science/pith/L7KF7M5MQR4VEYT7KUY2CFRFAC/bundle.json","state":"https://pith.science/pith/L7KF7M5MQR4VEYT7KUY2CFRFAC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L7KF7M5MQR4VEYT7KUY2CFRFAC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:L7KF7M5MQR4VEYT7KUY2CFRFAC","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":"183e38dc9d2a9bd525c4e0385d1c8c132340b4a79425fca90d15ee3effeee84c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-08-15T08:23:26Z","title_canon_sha256":"9a53fda2c87c56b7666a8b737900e280b76387309804f4ed1554febaf07d770f"},"schema_version":"1.0","source":{"id":"1708.04428","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.04428","created_at":"2026-05-18T00:23:23Z"},{"alias_kind":"arxiv_version","alias_value":"1708.04428v2","created_at":"2026-05-18T00:23:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.04428","created_at":"2026-05-18T00:23:23Z"},{"alias_kind":"pith_short_12","alias_value":"L7KF7M5MQR4V","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_16","alias_value":"L7KF7M5MQR4VEYT7","created_at":"2026-05-18T12:31:28Z"},{"alias_kind":"pith_short_8","alias_value":"L7KF7M5M","created_at":"2026-05-18T12:31:28Z"}],"graph_snapshots":[{"event_id":"sha256:7c9a56af633dd206438e61cd2b4f52f06d14b32239bcd4e02e89bccf782c0898","target":"graph","created_at":"2026-05-18T00:23:23Z","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 construct a monoidal category $\\mathscr{C}_{w,v}$ which categorifies the doubly-invariant algebra $^{N'(w)}\\mathbb{C}[N]^{N(v)}$ associated with Weyl group elements $w$ and $v$. It gives, after a localization, the coordinate algebra $\\mathbb{C}[\\mathcal{R}_{w,v}]$ of the open Richardson variety associated with $w$ and $v$. The category $\\mathscr{C}_{w,v}$ is realized as a subcategory of the graded module category of a quiver Hecke algebra $R$. When $v= \\mathrm{id}$, $\\mathscr{C}_{w,v}$ is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent c","authors_text":"Euiyong Park, Masaki Kashiwara, MyungHo Kim, Se-Jin Oh","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-08-15T08:23:26Z","title":"Monoidal categories associated with strata of flag manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.04428","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:de5e4165a19c317d96f0319695d585c81bc5a588dd308ba8dd93e405cad5be44","target":"record","created_at":"2026-05-18T00:23:23Z","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":"183e38dc9d2a9bd525c4e0385d1c8c132340b4a79425fca90d15ee3effeee84c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-08-15T08:23:26Z","title_canon_sha256":"9a53fda2c87c56b7666a8b737900e280b76387309804f4ed1554febaf07d770f"},"schema_version":"1.0","source":{"id":"1708.04428","kind":"arxiv","version":2}},"canonical_sha256":"5fd45fb3ac847952627f5531a1162500ba81cd188e6883a51c5dac3019b70a73","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5fd45fb3ac847952627f5531a1162500ba81cd188e6883a51c5dac3019b70a73","first_computed_at":"2026-05-18T00:23:23.156248Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:23:23.156248Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xlDM5oFVaALh+eREcjr5iWMlZpC1vLbBNY0pLsyhC4WSEO65g6pgf/bj+pHBYsaSwPu9m4VSS2FDWmvc0vFCDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:23:23.156958Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.04428","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de5e4165a19c317d96f0319695d585c81bc5a588dd308ba8dd93e405cad5be44","sha256:7c9a56af633dd206438e61cd2b4f52f06d14b32239bcd4e02e89bccf782c0898"],"state_sha256":"bdb0cb0468367285c13adc71ceacbf41623f552e5845e4b6b75010c8a468a28e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MRJtx1uwBgFfS+aTqITtwOGTC3788Tua8OyklMiysCEq7kgvB6SvDiJTlzc9EuxHzMBHGIdaofcrPMYsMG8VCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T17:03:08.318522Z","bundle_sha256":"0c66757cbc6effff96290ee83032d05e1e2b40f857beea8c6226bd92dc92f0b8"}}