{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:VQWTGG4JNLQWUVDH6ZH6EJZNZJ","short_pith_number":"pith:VQWTGG4J","canonical_record":{"source":{"id":"1803.03602","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-03-09T17:14:55Z","cross_cats_sorted":[],"title_canon_sha256":"17ef823bc54f934eb7da0882edb6599e3321d7477c8fe6caf0efebe87f4b56bc","abstract_canon_sha256":"c0266a055649cee5863f920c7eefcd001a2ab09dfbcb5f01dfaf863e0cd4216f"},"schema_version":"1.0"},"canonical_sha256":"ac2d331b896ae16a5467f64fe2272dca68fd6fd392fb9efe781e5ffdad2dd654","source":{"kind":"arxiv","id":"1803.03602","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.03602","created_at":"2026-05-17T23:59:58Z"},{"alias_kind":"arxiv_version","alias_value":"1803.03602v2","created_at":"2026-05-17T23:59:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.03602","created_at":"2026-05-17T23:59:58Z"},{"alias_kind":"pith_short_12","alias_value":"VQWTGG4JNLQW","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VQWTGG4JNLQWUVDH","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VQWTGG4J","created_at":"2026-05-18T12:32:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:VQWTGG4JNLQWUVDH6ZH6EJZNZJ","target":"record","payload":{"canonical_record":{"source":{"id":"1803.03602","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-03-09T17:14:55Z","cross_cats_sorted":[],"title_canon_sha256":"17ef823bc54f934eb7da0882edb6599e3321d7477c8fe6caf0efebe87f4b56bc","abstract_canon_sha256":"c0266a055649cee5863f920c7eefcd001a2ab09dfbcb5f01dfaf863e0cd4216f"},"schema_version":"1.0"},"canonical_sha256":"ac2d331b896ae16a5467f64fe2272dca68fd6fd392fb9efe781e5ffdad2dd654","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:58.095024Z","signature_b64":"Fu9X+wQTZ0p2P++WlgkRlGn5Tnsjz81dDWgi/AeCjQQZUDKmD5IPqhM7Pn+jhxz9nh6tEFmLXGOHJjzX93eoCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac2d331b896ae16a5467f64fe2272dca68fd6fd392fb9efe781e5ffdad2dd654","last_reissued_at":"2026-05-17T23:59:58.094425Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:58.094425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1803.03602","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":"2026-05-17T23:59:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B4lMLGI+pcpT7OgSMQgtMwKuKBWpZxjwxvSyfR+VltT1yhL2U4+7ojB4+odTF4d8Bmntp7/HtLSfT/qqAq09Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T09:53:34.895520Z"},"content_sha256":"c429a4f391adf1bef884938a33d4848d4fecc993b96199686d562e299bcfaf9a","schema_version":"1.0","event_id":"sha256:c429a4f391adf1bef884938a33d4848d4fecc993b96199686d562e299bcfaf9a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:VQWTGG4JNLQWUVDH6ZH6EJZNZJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weyl's polarization theorem in positive characteristic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Harm Derksen, Visu Makam","submitted_at":"2018-03-09T17:14:55Z","abstract_excerpt":"Let $V$ be an $n$-dimensional algebraic representation over an algebraically closed field $K$ of a group $G$. For $m > 0$, we study the invariant rings $K[V^{ m}]^G$ for the diagonal action of $G$ on $V^m$. In characteristic zero, a theorem of Weyl tells us that we can obtain all the invariants in $K[V^m]^G$ by the process of polarization and restitution from $K[V^n]^G$. In particular, this means that if $K[V^n]^G$ is generated in degree $\\leq d$, then so is $K[V^m]^G$ no matter how large $m$ is.\n  There are several explicit counterexamples to Weyl's theorem in positive characteristic. However"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.03602","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":"2026-05-17T23:59:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HiB8zgBHKTwXdFQRT4lIeN/Wqj/hRbdlSDaGPrydnMRP8iMJedaA6z7lqhebVR/oilWPp//rh3rgyO6rlR6wDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T09:53:34.895863Z"},"content_sha256":"a0b36e1c7417bb284ba943fdbd50cc2b73b91c5a05e46e8d9fff8fec23a83ec0","schema_version":"1.0","event_id":"sha256:a0b36e1c7417bb284ba943fdbd50cc2b73b91c5a05e46e8d9fff8fec23a83ec0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VQWTGG4JNLQWUVDH6ZH6EJZNZJ/bundle.json","state_url":"https://pith.science/pith/VQWTGG4JNLQWUVDH6ZH6EJZNZJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VQWTGG4JNLQWUVDH6ZH6EJZNZJ/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":"2026-07-02T09:53:34Z","links":{"resolver":"https://pith.science/pith/VQWTGG4JNLQWUVDH6ZH6EJZNZJ","bundle":"https://pith.science/pith/VQWTGG4JNLQWUVDH6ZH6EJZNZJ/bundle.json","state":"https://pith.science/pith/VQWTGG4JNLQWUVDH6ZH6EJZNZJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VQWTGG4JNLQWUVDH6ZH6EJZNZJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:VQWTGG4JNLQWUVDH6ZH6EJZNZJ","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":"c0266a055649cee5863f920c7eefcd001a2ab09dfbcb5f01dfaf863e0cd4216f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-03-09T17:14:55Z","title_canon_sha256":"17ef823bc54f934eb7da0882edb6599e3321d7477c8fe6caf0efebe87f4b56bc"},"schema_version":"1.0","source":{"id":"1803.03602","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1803.03602","created_at":"2026-05-17T23:59:58Z"},{"alias_kind":"arxiv_version","alias_value":"1803.03602v2","created_at":"2026-05-17T23:59:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.03602","created_at":"2026-05-17T23:59:58Z"},{"alias_kind":"pith_short_12","alias_value":"VQWTGG4JNLQW","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_16","alias_value":"VQWTGG4JNLQWUVDH","created_at":"2026-05-18T12:32:59Z"},{"alias_kind":"pith_short_8","alias_value":"VQWTGG4J","created_at":"2026-05-18T12:32:59Z"}],"graph_snapshots":[{"event_id":"sha256:a0b36e1c7417bb284ba943fdbd50cc2b73b91c5a05e46e8d9fff8fec23a83ec0","target":"graph","created_at":"2026-05-17T23:59:58Z","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 $V$ be an $n$-dimensional algebraic representation over an algebraically closed field $K$ of a group $G$. For $m > 0$, we study the invariant rings $K[V^{ m}]^G$ for the diagonal action of $G$ on $V^m$. In characteristic zero, a theorem of Weyl tells us that we can obtain all the invariants in $K[V^m]^G$ by the process of polarization and restitution from $K[V^n]^G$. In particular, this means that if $K[V^n]^G$ is generated in degree $\\leq d$, then so is $K[V^m]^G$ no matter how large $m$ is.\n  There are several explicit counterexamples to Weyl's theorem in positive characteristic. However","authors_text":"Harm Derksen, Visu Makam","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-03-09T17:14:55Z","title":"Weyl's polarization theorem in positive characteristic"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.03602","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:c429a4f391adf1bef884938a33d4848d4fecc993b96199686d562e299bcfaf9a","target":"record","created_at":"2026-05-17T23:59:58Z","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":"c0266a055649cee5863f920c7eefcd001a2ab09dfbcb5f01dfaf863e0cd4216f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2018-03-09T17:14:55Z","title_canon_sha256":"17ef823bc54f934eb7da0882edb6599e3321d7477c8fe6caf0efebe87f4b56bc"},"schema_version":"1.0","source":{"id":"1803.03602","kind":"arxiv","version":2}},"canonical_sha256":"ac2d331b896ae16a5467f64fe2272dca68fd6fd392fb9efe781e5ffdad2dd654","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac2d331b896ae16a5467f64fe2272dca68fd6fd392fb9efe781e5ffdad2dd654","first_computed_at":"2026-05-17T23:59:58.094425Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:58.094425Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Fu9X+wQTZ0p2P++WlgkRlGn5Tnsjz81dDWgi/AeCjQQZUDKmD5IPqhM7Pn+jhxz9nh6tEFmLXGOHJjzX93eoCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:58.095024Z","signed_message":"canonical_sha256_bytes"},"source_id":"1803.03602","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c429a4f391adf1bef884938a33d4848d4fecc993b96199686d562e299bcfaf9a","sha256:a0b36e1c7417bb284ba943fdbd50cc2b73b91c5a05e46e8d9fff8fec23a83ec0"],"state_sha256":"7d87ea1dfe8a2b52f737c0335f696e6653ec99bad164288f6ca04f006048fc87"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jcKenWnLWzDk4Dc8yL0CzBI5V84ONX3DNc+EcIF0v+F8yCGO8f2TTJpQmP5JFv7B0ljOk3fTnQPRucez5h6ACA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T09:53:34.897739Z","bundle_sha256":"8f44972fbc10aabc2e2a2899e4b756306a50de565b26e9b46f2debd3d32fce3f"}}