{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:XQ7XJWQYY6EKEN4IDEYI3UO3R2","short_pith_number":"pith:XQ7XJWQY","canonical_record":{"source":{"id":"1704.02405","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-04-08T00:03:36Z","cross_cats_sorted":[],"title_canon_sha256":"539e59e352517993221d136e73004c7c3e1cd838c53be7e217823637ea25d4d2","abstract_canon_sha256":"95fe666c412d8af40d6ee50e1f93d9f556c4c069f86aa48a45ffa31585327bc7"},"schema_version":"1.0"},"canonical_sha256":"bc3f74da18c788a2378819308dd1db8eaf517ebe0d356acf608053c3ccde3eed","source":{"kind":"arxiv","id":"1704.02405","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.02405","created_at":"2026-05-18T00:46:45Z"},{"alias_kind":"arxiv_version","alias_value":"1704.02405v1","created_at":"2026-05-18T00:46:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.02405","created_at":"2026-05-18T00:46:45Z"},{"alias_kind":"pith_short_12","alias_value":"XQ7XJWQYY6EK","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"XQ7XJWQYY6EKEN4I","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"XQ7XJWQY","created_at":"2026-05-18T12:31:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:XQ7XJWQYY6EKEN4IDEYI3UO3R2","target":"record","payload":{"canonical_record":{"source":{"id":"1704.02405","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-04-08T00:03:36Z","cross_cats_sorted":[],"title_canon_sha256":"539e59e352517993221d136e73004c7c3e1cd838c53be7e217823637ea25d4d2","abstract_canon_sha256":"95fe666c412d8af40d6ee50e1f93d9f556c4c069f86aa48a45ffa31585327bc7"},"schema_version":"1.0"},"canonical_sha256":"bc3f74da18c788a2378819308dd1db8eaf517ebe0d356acf608053c3ccde3eed","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:45.686544Z","signature_b64":"02TnvOm4OLP8ctRoivFUKUeKD4mO7KhFK1O1ROj2dqg8uFudsXrzuljDhn/v+NaGG0DMoPFQYeO7oDXRvjyACQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bc3f74da18c788a2378819308dd1db8eaf517ebe0d356acf608053c3ccde3eed","last_reissued_at":"2026-05-18T00:46:45.685871Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:45.685871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1704.02405","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:46:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m18d45+MpoiMlF+7sqEWLqN0TNQVtxN6esPZVydu2zScEwqC/T1XQEq71YMpAs6VYHqEoCreUrlds0SS7lEjCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T20:29:27.089144Z"},"content_sha256":"685e28d0b685088288463277a99cbdd8cee8612ddcaed8f0f10acb276e574311","schema_version":"1.0","event_id":"sha256:685e28d0b685088288463277a99cbdd8cee8612ddcaed8f0f10acb276e574311"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:XQ7XJWQYY6EKEN4IDEYI3UO3R2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Polynomially and Infinitesimally Injective Modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Haralampos Geranios, Stephen Donkin","submitted_at":"2017-04-08T00:03:36Z","abstract_excerpt":"The injective polynomial modules for a general linear group $G$ of degree $n$ are labelled by the partitions with at most $n$ parts. Working over an algebraically closed field of characteristic $p$, we consider the question of which partitions correspond to polynomially injective modules that are also injective as modules for the restricted enveloping algebra of the Lie algebra of $G$. The question is related to the \"index of divisibility\" of a polynomial module in general, and an explicit answer is given for $n=2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.02405","kind":"arxiv","version":1},"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":"2026-05-18T00:46:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"25W1Pd0wmAqTKBCSlLyLnf6PQ+7sM9IteAhfD1AAy3A9lwnPtf4CdM1ruZlzXLIJkS4cOxY0R3clz3dBcVM7BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T20:29:27.089624Z"},"content_sha256":"71104cd756b1906359105c0cb86842ce4af18a3605e034c528a34f07c513c2fe","schema_version":"1.0","event_id":"sha256:71104cd756b1906359105c0cb86842ce4af18a3605e034c528a34f07c513c2fe"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XQ7XJWQYY6EKEN4IDEYI3UO3R2/bundle.json","state_url":"https://pith.science/pith/XQ7XJWQYY6EKEN4IDEYI3UO3R2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XQ7XJWQYY6EKEN4IDEYI3UO3R2/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":"2026-07-01T20:29:27Z","links":{"resolver":"https://pith.science/pith/XQ7XJWQYY6EKEN4IDEYI3UO3R2","bundle":"https://pith.science/pith/XQ7XJWQYY6EKEN4IDEYI3UO3R2/bundle.json","state":"https://pith.science/pith/XQ7XJWQYY6EKEN4IDEYI3UO3R2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XQ7XJWQYY6EKEN4IDEYI3UO3R2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:XQ7XJWQYY6EKEN4IDEYI3UO3R2","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":"95fe666c412d8af40d6ee50e1f93d9f556c4c069f86aa48a45ffa31585327bc7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-04-08T00:03:36Z","title_canon_sha256":"539e59e352517993221d136e73004c7c3e1cd838c53be7e217823637ea25d4d2"},"schema_version":"1.0","source":{"id":"1704.02405","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.02405","created_at":"2026-05-18T00:46:45Z"},{"alias_kind":"arxiv_version","alias_value":"1704.02405v1","created_at":"2026-05-18T00:46:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.02405","created_at":"2026-05-18T00:46:45Z"},{"alias_kind":"pith_short_12","alias_value":"XQ7XJWQYY6EK","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"XQ7XJWQYY6EKEN4I","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"XQ7XJWQY","created_at":"2026-05-18T12:31:56Z"}],"graph_snapshots":[{"event_id":"sha256:71104cd756b1906359105c0cb86842ce4af18a3605e034c528a34f07c513c2fe","target":"graph","created_at":"2026-05-18T00:46:45Z","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":"The injective polynomial modules for a general linear group $G$ of degree $n$ are labelled by the partitions with at most $n$ parts. Working over an algebraically closed field of characteristic $p$, we consider the question of which partitions correspond to polynomially injective modules that are also injective as modules for the restricted enveloping algebra of the Lie algebra of $G$. The question is related to the \"index of divisibility\" of a polynomial module in general, and an explicit answer is given for $n=2$.","authors_text":"Haralampos Geranios, Stephen Donkin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-04-08T00:03:36Z","title":"Polynomially and Infinitesimally Injective Modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.02405","kind":"arxiv","version":1},"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":"sha256:685e28d0b685088288463277a99cbdd8cee8612ddcaed8f0f10acb276e574311","target":"record","created_at":"2026-05-18T00:46:45Z","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":"95fe666c412d8af40d6ee50e1f93d9f556c4c069f86aa48a45ffa31585327bc7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-04-08T00:03:36Z","title_canon_sha256":"539e59e352517993221d136e73004c7c3e1cd838c53be7e217823637ea25d4d2"},"schema_version":"1.0","source":{"id":"1704.02405","kind":"arxiv","version":1}},"canonical_sha256":"bc3f74da18c788a2378819308dd1db8eaf517ebe0d356acf608053c3ccde3eed","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bc3f74da18c788a2378819308dd1db8eaf517ebe0d356acf608053c3ccde3eed","first_computed_at":"2026-05-18T00:46:45.685871Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:46:45.685871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"02TnvOm4OLP8ctRoivFUKUeKD4mO7KhFK1O1ROj2dqg8uFudsXrzuljDhn/v+NaGG0DMoPFQYeO7oDXRvjyACQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:46:45.686544Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.02405","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:685e28d0b685088288463277a99cbdd8cee8612ddcaed8f0f10acb276e574311","sha256:71104cd756b1906359105c0cb86842ce4af18a3605e034c528a34f07c513c2fe"],"state_sha256":"c744c8e7f279fe69c0d7ecbb8b9c72d269d6404128cd7d1399bf6040f91a2346"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RUjsutB+NcEx+DWVeT/dTHrn7bXr13ZjmwTXAthBnuzxGWZdYgsFViPJDyTrtf+NLAbt1XRUDNjAMSSaaDcQAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T20:29:27.092418Z","bundle_sha256":"69b3d38246936a0bb5ab48189be6c5820cc9ae5f8500cb28ac6f877d4fdd41aa"}}