{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:5RR5ILDRIFZ4JEABNRS5BETVZQ","short_pith_number":"pith:5RR5ILDR","canonical_record":{"source":{"id":"1209.5122","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-09-23T22:45:41Z","cross_cats_sorted":["math.AG","math.RT"],"title_canon_sha256":"8156e81d672d07a025bbaa42d00ea91f5082057e1e02582d4e56401d570e5d89","abstract_canon_sha256":"172eabc431c907a04300ea25c0b6466c80152bcd14030c999a6ed1b71473773f"},"schema_version":"1.0"},"canonical_sha256":"ec63d42c714173c490016c65d09275cc030c36d9ebd239f8a72fc80033e608b0","source":{"kind":"arxiv","id":"1209.5122","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.5122","created_at":"2026-05-18T03:44:59Z"},{"alias_kind":"arxiv_version","alias_value":"1209.5122v1","created_at":"2026-05-18T03:44:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.5122","created_at":"2026-05-18T03:44:59Z"},{"alias_kind":"pith_short_12","alias_value":"5RR5ILDRIFZ4","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"5RR5ILDRIFZ4JEAB","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"5RR5ILDR","created_at":"2026-05-18T12:26:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:5RR5ILDRIFZ4JEABNRS5BETVZQ","target":"record","payload":{"canonical_record":{"source":{"id":"1209.5122","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-09-23T22:45:41Z","cross_cats_sorted":["math.AG","math.RT"],"title_canon_sha256":"8156e81d672d07a025bbaa42d00ea91f5082057e1e02582d4e56401d570e5d89","abstract_canon_sha256":"172eabc431c907a04300ea25c0b6466c80152bcd14030c999a6ed1b71473773f"},"schema_version":"1.0"},"canonical_sha256":"ec63d42c714173c490016c65d09275cc030c36d9ebd239f8a72fc80033e608b0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:59.161472Z","signature_b64":"oIpsVE97Y4OsEC0w/2zg06ZqLDOxpO9/caMGSZWmqNfu+sc169Po4z7lUgu8v6m3ExsTmZSz2FMB9n2pUvCPBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ec63d42c714173c490016c65d09275cc030c36d9ebd239f8a72fc80033e608b0","last_reissued_at":"2026-05-18T03:44:59.160879Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:59.160879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.5122","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":"2026-05-18T03:44:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QMHljDPkBS9CjRK9kocG+2a9TLUqz5boDDMnZ4SDS4o62nAlRcViYhnhvtAZbijTCE2t6x/Tv0hWlJDjVCLtCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T21:56:57.940832Z"},"content_sha256":"e146da9cca1bf576b01ff5a17fcc6a05ad21bcdc0398b60d4459b57cd31cf75e","schema_version":"1.0","event_id":"sha256:e146da9cca1bf576b01ff5a17fcc6a05ad21bcdc0398b60d4459b57cd31cf75e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:5RR5ILDRIFZ4JEABNRS5BETVZQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Introduction to twisted commutative algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.RT"],"primary_cat":"math.AC","authors_text":"Andrew Snowden, Steven V Sam","submitted_at":"2012-09-23T22:45:41Z","abstract_excerpt":"This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate rings of determinantal varieties, Segre-Veronese embeddings, and Grassmannians. The article is meant to serve as a gentle introduction to the papers of the two authors on the subject, and also to point out some literature in which these algebras appear. The first part reviews the representation theory of the symmetric groups and general linear groups. The sec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5122","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":"2026-05-18T03:44:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5wVeHZUFX5gOyA/HRMZ0EJdhsiuuogZquhh9CVoZGVoEll10qudXeV2eDQR7xh2nRIOEg5dOq3fclhjLMRr1Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T21:56:57.941181Z"},"content_sha256":"f70675390d874433b9f7fd92eff6999d0a69d22292fa8c0563acf6919a4fef45","schema_version":"1.0","event_id":"sha256:f70675390d874433b9f7fd92eff6999d0a69d22292fa8c0563acf6919a4fef45"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5RR5ILDRIFZ4JEABNRS5BETVZQ/bundle.json","state_url":"https://pith.science/pith/5RR5ILDRIFZ4JEABNRS5BETVZQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5RR5ILDRIFZ4JEABNRS5BETVZQ/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":"2026-05-30T21:56:57Z","links":{"resolver":"https://pith.science/pith/5RR5ILDRIFZ4JEABNRS5BETVZQ","bundle":"https://pith.science/pith/5RR5ILDRIFZ4JEABNRS5BETVZQ/bundle.json","state":"https://pith.science/pith/5RR5ILDRIFZ4JEABNRS5BETVZQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5RR5ILDRIFZ4JEABNRS5BETVZQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:5RR5ILDRIFZ4JEABNRS5BETVZQ","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":"172eabc431c907a04300ea25c0b6466c80152bcd14030c999a6ed1b71473773f","cross_cats_sorted":["math.AG","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-09-23T22:45:41Z","title_canon_sha256":"8156e81d672d07a025bbaa42d00ea91f5082057e1e02582d4e56401d570e5d89"},"schema_version":"1.0","source":{"id":"1209.5122","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.5122","created_at":"2026-05-18T03:44:59Z"},{"alias_kind":"arxiv_version","alias_value":"1209.5122v1","created_at":"2026-05-18T03:44:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.5122","created_at":"2026-05-18T03:44:59Z"},{"alias_kind":"pith_short_12","alias_value":"5RR5ILDRIFZ4","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"5RR5ILDRIFZ4JEAB","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"5RR5ILDR","created_at":"2026-05-18T12:26:56Z"}],"graph_snapshots":[{"event_id":"sha256:f70675390d874433b9f7fd92eff6999d0a69d22292fa8c0563acf6919a4fef45","target":"graph","created_at":"2026-05-18T03:44:59Z","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":"This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate rings of determinantal varieties, Segre-Veronese embeddings, and Grassmannians. The article is meant to serve as a gentle introduction to the papers of the two authors on the subject, and also to point out some literature in which these algebras appear. The first part reviews the representation theory of the symmetric groups and general linear groups. The sec","authors_text":"Andrew Snowden, Steven V Sam","cross_cats":["math.AG","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-09-23T22:45:41Z","title":"Introduction to twisted commutative algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5122","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:e146da9cca1bf576b01ff5a17fcc6a05ad21bcdc0398b60d4459b57cd31cf75e","target":"record","created_at":"2026-05-18T03:44:59Z","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":"172eabc431c907a04300ea25c0b6466c80152bcd14030c999a6ed1b71473773f","cross_cats_sorted":["math.AG","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2012-09-23T22:45:41Z","title_canon_sha256":"8156e81d672d07a025bbaa42d00ea91f5082057e1e02582d4e56401d570e5d89"},"schema_version":"1.0","source":{"id":"1209.5122","kind":"arxiv","version":1}},"canonical_sha256":"ec63d42c714173c490016c65d09275cc030c36d9ebd239f8a72fc80033e608b0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ec63d42c714173c490016c65d09275cc030c36d9ebd239f8a72fc80033e608b0","first_computed_at":"2026-05-18T03:44:59.160879Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:44:59.160879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"oIpsVE97Y4OsEC0w/2zg06ZqLDOxpO9/caMGSZWmqNfu+sc169Po4z7lUgu8v6m3ExsTmZSz2FMB9n2pUvCPBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:44:59.161472Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.5122","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e146da9cca1bf576b01ff5a17fcc6a05ad21bcdc0398b60d4459b57cd31cf75e","sha256:f70675390d874433b9f7fd92eff6999d0a69d22292fa8c0563acf6919a4fef45"],"state_sha256":"2baa3f54e87336136761db17ff75dfc1c009e4049a0b903095e41e2de87f404b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u+anFOW7tWj718+ylHI4s7Lds4qgHLsxwr4WrFIUGJP/JPjY1rTTjIY0x8D+cySY4BIkioVnApBMxoKHZe5FCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T21:56:57.943224Z","bundle_sha256":"e50a780c630c5d1bf141fe8b07b32aa1245341313f62da566bd99029a3b8009b"}}