{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:RGXUVDP4SL2QT4PQ45CQUES67R","short_pith_number":"pith:RGXUVDP4","canonical_record":{"source":{"id":"1006.4762","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-06-24T11:55:11Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"08c92ccf62b6b04cab40810a5114834a1ce5a8505e5ace42ad587371c677b6ad","abstract_canon_sha256":"74fe2c289f31ff868a8ce94cf58dab9068a5f2f02030a667117eac8125c3f84d"},"schema_version":"1.0"},"canonical_sha256":"89af4a8dfc92f509f1f0e7450a125efc4938e7765506846a7887e895f63b0087","source":{"kind":"arxiv","id":"1006.4762","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.4762","created_at":"2026-05-18T04:25:11Z"},{"alias_kind":"arxiv_version","alias_value":"1006.4762v2","created_at":"2026-05-18T04:25:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.4762","created_at":"2026-05-18T04:25:11Z"},{"alias_kind":"pith_short_12","alias_value":"RGXUVDP4SL2Q","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"RGXUVDP4SL2QT4PQ","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"RGXUVDP4","created_at":"2026-05-18T12:26:13Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:RGXUVDP4SL2QT4PQ45CQUES67R","target":"record","payload":{"canonical_record":{"source":{"id":"1006.4762","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-06-24T11:55:11Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"08c92ccf62b6b04cab40810a5114834a1ce5a8505e5ace42ad587371c677b6ad","abstract_canon_sha256":"74fe2c289f31ff868a8ce94cf58dab9068a5f2f02030a667117eac8125c3f84d"},"schema_version":"1.0"},"canonical_sha256":"89af4a8dfc92f509f1f0e7450a125efc4938e7765506846a7887e895f63b0087","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:11.341901Z","signature_b64":"d0A2+GIslrfNpPkqPr4JFf5pzqLOIrfI4g0fXcZg3Ng0Ng95fdkuvdkA3DtkYc+nmtkyZu7arTrZyydIuSboCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"89af4a8dfc92f509f1f0e7450a125efc4938e7765506846a7887e895f63b0087","last_reissued_at":"2026-05-18T04:25:11.341285Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:11.341285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1006.4762","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":"2026-05-18T04:25:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AzgGjn1gbW2Q3aWgT/eo1esel2oVxmlyvlyyqtPF4Rlt3EqKgPWEnMzQVXYjkHv4TBaqHq3iOcRJM5gjdlEWAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T08:09:43.401294Z"},"content_sha256":"64b4b56420231d69ad41a242d11c4f3cdb48a448c52b97a526cba42791baa83b","schema_version":"1.0","event_id":"sha256:64b4b56420231d69ad41a242d11c4f3cdb48a448c52b97a526cba42791baa83b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:RGXUVDP4SL2QT4PQ45CQUES67R","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some complete intersection symplectic quotients in positive characteristic: invariants of a vector and a covector","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"C\\'edric Bonnaf\\'e (LM-Besan\\c{c}on), G. Kemper","submitted_at":"2010-06-24T11:55:11Z","abstract_excerpt":"Given a linear action of a group $G$ on a $K$-vector space $V$, we consider the invariant ring $K[V \\oplus V^*]^G$, where $V^*$ is the dual space. We are particularly interested in the case where $V =\\gfq^n$ and $G$ is the group $U_n$ of all upper unipotent matrices or the group $B_n$ of all upper triangular matrices in $\\GL_n(\\gfq)$. In fact, we determine $\\gfq[V \\oplus V^*]^G$ for $G = U_n$ and $G =B_n$. The result is a complete intersection for all values of $n$ and $q$. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.4762","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":"2026-05-18T04:25:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kO9UXEdddxnh4HwCaGm4u6aNJR+3evkIRcH7mJhX6AqANBPBVICV4hYoKQza8LM97Spre6NoNN+rHSJmcmRdDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T08:09:43.401657Z"},"content_sha256":"c7648fb9e942ce42bc46b2ccf7e9ff13e06fed7e51413b57789a81a350d20f32","schema_version":"1.0","event_id":"sha256:c7648fb9e942ce42bc46b2ccf7e9ff13e06fed7e51413b57789a81a350d20f32"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RGXUVDP4SL2QT4PQ45CQUES67R/bundle.json","state_url":"https://pith.science/pith/RGXUVDP4SL2QT4PQ45CQUES67R/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RGXUVDP4SL2QT4PQ45CQUES67R/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":"2026-05-28T08:09:43Z","links":{"resolver":"https://pith.science/pith/RGXUVDP4SL2QT4PQ45CQUES67R","bundle":"https://pith.science/pith/RGXUVDP4SL2QT4PQ45CQUES67R/bundle.json","state":"https://pith.science/pith/RGXUVDP4SL2QT4PQ45CQUES67R/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RGXUVDP4SL2QT4PQ45CQUES67R/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:RGXUVDP4SL2QT4PQ45CQUES67R","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":"74fe2c289f31ff868a8ce94cf58dab9068a5f2f02030a667117eac8125c3f84d","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-06-24T11:55:11Z","title_canon_sha256":"08c92ccf62b6b04cab40810a5114834a1ce5a8505e5ace42ad587371c677b6ad"},"schema_version":"1.0","source":{"id":"1006.4762","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.4762","created_at":"2026-05-18T04:25:11Z"},{"alias_kind":"arxiv_version","alias_value":"1006.4762v2","created_at":"2026-05-18T04:25:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.4762","created_at":"2026-05-18T04:25:11Z"},{"alias_kind":"pith_short_12","alias_value":"RGXUVDP4SL2Q","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_16","alias_value":"RGXUVDP4SL2QT4PQ","created_at":"2026-05-18T12:26:13Z"},{"alias_kind":"pith_short_8","alias_value":"RGXUVDP4","created_at":"2026-05-18T12:26:13Z"}],"graph_snapshots":[{"event_id":"sha256:c7648fb9e942ce42bc46b2ccf7e9ff13e06fed7e51413b57789a81a350d20f32","target":"graph","created_at":"2026-05-18T04:25:11Z","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":"Given a linear action of a group $G$ on a $K$-vector space $V$, we consider the invariant ring $K[V \\oplus V^*]^G$, where $V^*$ is the dual space. We are particularly interested in the case where $V =\\gfq^n$ and $G$ is the group $U_n$ of all upper unipotent matrices or the group $B_n$ of all upper triangular matrices in $\\GL_n(\\gfq)$. In fact, we determine $\\gfq[V \\oplus V^*]^G$ for $G = U_n$ and $G =B_n$. The result is a complete intersection for all values of $n$ and $q$. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list o","authors_text":"C\\'edric Bonnaf\\'e (LM-Besan\\c{c}on), G. Kemper","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-06-24T11:55:11Z","title":"Some complete intersection symplectic quotients in positive characteristic: invariants of a vector and a covector"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.4762","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":"sha256:64b4b56420231d69ad41a242d11c4f3cdb48a448c52b97a526cba42791baa83b","target":"record","created_at":"2026-05-18T04:25:11Z","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":"74fe2c289f31ff868a8ce94cf58dab9068a5f2f02030a667117eac8125c3f84d","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-06-24T11:55:11Z","title_canon_sha256":"08c92ccf62b6b04cab40810a5114834a1ce5a8505e5ace42ad587371c677b6ad"},"schema_version":"1.0","source":{"id":"1006.4762","kind":"arxiv","version":2}},"canonical_sha256":"89af4a8dfc92f509f1f0e7450a125efc4938e7765506846a7887e895f63b0087","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"89af4a8dfc92f509f1f0e7450a125efc4938e7765506846a7887e895f63b0087","first_computed_at":"2026-05-18T04:25:11.341285Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:25:11.341285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d0A2+GIslrfNpPkqPr4JFf5pzqLOIrfI4g0fXcZg3Ng0Ng95fdkuvdkA3DtkYc+nmtkyZu7arTrZyydIuSboCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:25:11.341901Z","signed_message":"canonical_sha256_bytes"},"source_id":"1006.4762","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:64b4b56420231d69ad41a242d11c4f3cdb48a448c52b97a526cba42791baa83b","sha256:c7648fb9e942ce42bc46b2ccf7e9ff13e06fed7e51413b57789a81a350d20f32"],"state_sha256":"2bced9daf20a36c50d346c582f48654ab75c79c22c9f6de219eab85be7436b57"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tRvDfIdlD3hjheby7nK5NdqC6F4iIjgokGl1PINLss0QhjwmpxSB3fpe3LPGf6eHV1RAhVDlSBGUBrKnUY05Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T08:09:43.403639Z","bundle_sha256":"485191c5050ec0a08603f4e65995a9daa3190cf15f8aab1e9ec1a06429d9dcee"}}