{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:YOTAXXHOTS3DMMWNEXC2EGEL7Z","short_pith_number":"pith:YOTAXXHO","canonical_record":{"source":{"id":"1209.5117","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-23T21:52:04Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ca1acea38c3c2cf223124aace726b0f85f5672e78cda33bc563f1bcbaeffefcd","abstract_canon_sha256":"049af4e96a25362c8828b37c90e665813ed416279328c14eca0554fa210579f5"},"schema_version":"1.0"},"canonical_sha256":"c3a60bdcee9cb63632cd25c5a2188bfe577ac373f067c36343cac444aa3b0a2e","source":{"kind":"arxiv","id":"1209.5117","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.5117","created_at":"2026-05-18T03:44:59Z"},{"alias_kind":"arxiv_version","alias_value":"1209.5117v1","created_at":"2026-05-18T03:44:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.5117","created_at":"2026-05-18T03:44:59Z"},{"alias_kind":"pith_short_12","alias_value":"YOTAXXHOTS3D","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"YOTAXXHOTS3DMMWN","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"YOTAXXHO","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:YOTAXXHOTS3DMMWNEXC2EGEL7Z","target":"record","payload":{"canonical_record":{"source":{"id":"1209.5117","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-23T21:52:04Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ca1acea38c3c2cf223124aace726b0f85f5672e78cda33bc563f1bcbaeffefcd","abstract_canon_sha256":"049af4e96a25362c8828b37c90e665813ed416279328c14eca0554fa210579f5"},"schema_version":"1.0"},"canonical_sha256":"c3a60bdcee9cb63632cd25c5a2188bfe577ac373f067c36343cac444aa3b0a2e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:59.192343Z","signature_b64":"99TAb42FmFJKPcT4ibj5w9HKVprIrTPavF/GrgjQG8DcKJdc5Do7ErpQXYfA9XgcQ5iFvBUdssUw0iHCYkKECA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c3a60bdcee9cb63632cd25c5a2188bfe577ac373f067c36343cac444aa3b0a2e","last_reissued_at":"2026-05-18T03:44:59.191641Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:59.191641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.5117","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":"CmzgBn9No+yV8DEq+NfpWDQCV1itKjY8Cwg0LhIzDFgVpfEZ7cfv2bKpgwF2HaoUVC5/4s/SFo8fnUame6pfDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T15:25:15.715870Z"},"content_sha256":"f14d9a0cae608efa7224576fa3bd66d3ba7766498da04fa6cd7cea1b8353461c","schema_version":"1.0","event_id":"sha256:f14d9a0cae608efa7224576fa3bd66d3ba7766498da04fa6cd7cea1b8353461c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:YOTAXXHOTS3DMMWNEXC2EGEL7Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Invariant polynomial functions on tensors under the action of a product of orthogonal groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Lauren Kelly Williams","submitted_at":"2012-09-23T21:52:04Z","abstract_excerpt":"Let K be the product O(n_1) x O(n_2) x ... x O(n_r) of orthogonal groups. Let V the r-fold tensor product of defining representations of each orthogonal factor. We compute a stable formula for the dimension of the K-invariant algebra of degree d homogeneous polynomial functions on V. To accomplish this, we compute a formula for the number of matchings which commute with a fixed permutation. Finally, we provide formulas for the invariants and describe a bijection between a basis for the space of invariants and the isomorphism classes of certain r-regular graphs on d vertices, as well as a metho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5117","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":"xCbuNU8+uRiWwPRDqftE1Ow7E3/qEEf1D20tXnQ+4DYEMbjIguAYLnGrx73V2mbVh16cSKcw0BpS2J67h8LiCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T15:25:15.716280Z"},"content_sha256":"868f304a41044e74f13a50c8f0bf055883f3d4fba7031a052b1605ea3ffee025","schema_version":"1.0","event_id":"sha256:868f304a41044e74f13a50c8f0bf055883f3d4fba7031a052b1605ea3ffee025"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YOTAXXHOTS3DMMWNEXC2EGEL7Z/bundle.json","state_url":"https://pith.science/pith/YOTAXXHOTS3DMMWNEXC2EGEL7Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YOTAXXHOTS3DMMWNEXC2EGEL7Z/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":"2026-05-28T15:25:15Z","links":{"resolver":"https://pith.science/pith/YOTAXXHOTS3DMMWNEXC2EGEL7Z","bundle":"https://pith.science/pith/YOTAXXHOTS3DMMWNEXC2EGEL7Z/bundle.json","state":"https://pith.science/pith/YOTAXXHOTS3DMMWNEXC2EGEL7Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YOTAXXHOTS3DMMWNEXC2EGEL7Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:YOTAXXHOTS3DMMWNEXC2EGEL7Z","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":"049af4e96a25362c8828b37c90e665813ed416279328c14eca0554fa210579f5","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-23T21:52:04Z","title_canon_sha256":"ca1acea38c3c2cf223124aace726b0f85f5672e78cda33bc563f1bcbaeffefcd"},"schema_version":"1.0","source":{"id":"1209.5117","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.5117","created_at":"2026-05-18T03:44:59Z"},{"alias_kind":"arxiv_version","alias_value":"1209.5117v1","created_at":"2026-05-18T03:44:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.5117","created_at":"2026-05-18T03:44:59Z"},{"alias_kind":"pith_short_12","alias_value":"YOTAXXHOTS3D","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"YOTAXXHOTS3DMMWN","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"YOTAXXHO","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:868f304a41044e74f13a50c8f0bf055883f3d4fba7031a052b1605ea3ffee025","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":"Let K be the product O(n_1) x O(n_2) x ... x O(n_r) of orthogonal groups. Let V the r-fold tensor product of defining representations of each orthogonal factor. We compute a stable formula for the dimension of the K-invariant algebra of degree d homogeneous polynomial functions on V. To accomplish this, we compute a formula for the number of matchings which commute with a fixed permutation. Finally, we provide formulas for the invariants and describe a bijection between a basis for the space of invariants and the isomorphism classes of certain r-regular graphs on d vertices, as well as a metho","authors_text":"Lauren Kelly Williams","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-23T21:52:04Z","title":"Invariant polynomial functions on tensors under the action of a product of orthogonal groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.5117","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:f14d9a0cae608efa7224576fa3bd66d3ba7766498da04fa6cd7cea1b8353461c","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":"049af4e96a25362c8828b37c90e665813ed416279328c14eca0554fa210579f5","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-09-23T21:52:04Z","title_canon_sha256":"ca1acea38c3c2cf223124aace726b0f85f5672e78cda33bc563f1bcbaeffefcd"},"schema_version":"1.0","source":{"id":"1209.5117","kind":"arxiv","version":1}},"canonical_sha256":"c3a60bdcee9cb63632cd25c5a2188bfe577ac373f067c36343cac444aa3b0a2e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c3a60bdcee9cb63632cd25c5a2188bfe577ac373f067c36343cac444aa3b0a2e","first_computed_at":"2026-05-18T03:44:59.191641Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:44:59.191641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"99TAb42FmFJKPcT4ibj5w9HKVprIrTPavF/GrgjQG8DcKJdc5Do7ErpQXYfA9XgcQ5iFvBUdssUw0iHCYkKECA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:44:59.192343Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.5117","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f14d9a0cae608efa7224576fa3bd66d3ba7766498da04fa6cd7cea1b8353461c","sha256:868f304a41044e74f13a50c8f0bf055883f3d4fba7031a052b1605ea3ffee025"],"state_sha256":"cf2848b8b2ef250265ceaedc65701c016c2a6236d5badbcb9366d332004fb584"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eDZtWV7Z5MAYW1NQN5STPm8fvCgFn4rZoZ59uUjXcelJi5afckXQgE5uaL4fm/Zz4OuY9UEGDEUHD/kvBJgAAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T15:25:15.718606Z","bundle_sha256":"723058f05602c65c58b9fe361df299f0635aa30a4af4fa1052ac9a2f05106aab"}}