{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:2BR2URIHQHIYZJLZM2C45PR45Q","short_pith_number":"pith:2BR2URIH","canonical_record":{"source":{"id":"1908.02663","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-07T14:44:09Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"8e0f9d4d0e050e9629d70c172653734608b5e35ce17cc78b2df6d9ca70d17090","abstract_canon_sha256":"121a67bbba1172c43c6453a89eece9cf338630a1200999d9d61f590ce97c2745"},"schema_version":"1.0"},"canonical_sha256":"d063aa450781d18ca5796685cebe3cec1172e19eb8fadbd11ece67059274a3aa","source":{"kind":"arxiv","id":"1908.02663","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.02663","created_at":"2026-07-05T00:03:17Z"},{"alias_kind":"arxiv_version","alias_value":"1908.02663v2","created_at":"2026-07-05T00:03:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.02663","created_at":"2026-07-05T00:03:17Z"},{"alias_kind":"pith_short_12","alias_value":"2BR2URIHQHIY","created_at":"2026-07-05T00:03:17Z"},{"alias_kind":"pith_short_16","alias_value":"2BR2URIHQHIYZJLZ","created_at":"2026-07-05T00:03:17Z"},{"alias_kind":"pith_short_8","alias_value":"2BR2URIH","created_at":"2026-07-05T00:03:17Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:2BR2URIHQHIYZJLZM2C45PR45Q","target":"record","payload":{"canonical_record":{"source":{"id":"1908.02663","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-07T14:44:09Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"8e0f9d4d0e050e9629d70c172653734608b5e35ce17cc78b2df6d9ca70d17090","abstract_canon_sha256":"121a67bbba1172c43c6453a89eece9cf338630a1200999d9d61f590ce97c2745"},"schema_version":"1.0"},"canonical_sha256":"d063aa450781d18ca5796685cebe3cec1172e19eb8fadbd11ece67059274a3aa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:03:17.021683Z","signature_b64":"ivyKwZZUZsVLce4DLWfE1692WcO+nJs7ryDDq3ASQIoXp0JHSZmV424XzadZaELcXFaR58Pi9tqB0qtpf+44Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d063aa450781d18ca5796685cebe3cec1172e19eb8fadbd11ece67059274a3aa","last_reissued_at":"2026-07-05T00:03:17.021267Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:03:17.021267Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1908.02663","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":"2026-07-05T00:03:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5sFT/QLh7l0Z/gzScE1RRW8XrDuseR06qbVGqmkdfJ6IOOvJ9DI9jxvngF66XtIooeUfPOxc5jN+wG554spVCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T13:37:45.839033Z"},"content_sha256":"1eebd69cf114b54cf89f72338a78e5d493054f95504906b9e4321629bcc86b5a","schema_version":"1.0","event_id":"sha256:1eebd69cf114b54cf89f72338a78e5d493054f95504906b9e4321629bcc86b5a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:2BR2URIHQHIYZJLZM2C45PR45Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Invariant theory for coincidental complex reflection groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Anne V. Shepler, Eric Sommers, Victor Reiner","submitted_at":"2019-08-07T14:44:09Z","abstract_excerpt":"V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and speculated that it had a certain product formula involving the exponents of the group. We show that Molchanov's speculation is false in general but holds for all coincidental complex reflection groups when appropriately modified using exponents and co-exponents. These are the irreducible well-generated (i.e., duality) reflection groups with exponents forming an arithmetic progression and include many real r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.02663","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.02663/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T00:03:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8pQVJ6J5ZaH32GuAT0C9UCWxjCtEx+GM6NtXlLpZYovKbJrv3zd8CiRuVfi4XPi1IhPympf0VqqsIY4n4imtDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T13:37:45.839678Z"},"content_sha256":"c000907305b18cd78a6f7d4140f1d9125f48ff51f45bdf72dbe86c544709f520","schema_version":"1.0","event_id":"sha256:c000907305b18cd78a6f7d4140f1d9125f48ff51f45bdf72dbe86c544709f520"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2BR2URIHQHIYZJLZM2C45PR45Q/bundle.json","state_url":"https://pith.science/pith/2BR2URIHQHIYZJLZM2C45PR45Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2BR2URIHQHIYZJLZM2C45PR45Q/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":"2026-08-16T13:37:45Z","links":{"resolver":"https://pith.science/pith/2BR2URIHQHIYZJLZM2C45PR45Q","bundle":"https://pith.science/pith/2BR2URIHQHIYZJLZM2C45PR45Q/bundle.json","state":"https://pith.science/pith/2BR2URIHQHIYZJLZM2C45PR45Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2BR2URIHQHIYZJLZM2C45PR45Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:2BR2URIHQHIYZJLZM2C45PR45Q","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":"121a67bbba1172c43c6453a89eece9cf338630a1200999d9d61f590ce97c2745","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-07T14:44:09Z","title_canon_sha256":"8e0f9d4d0e050e9629d70c172653734608b5e35ce17cc78b2df6d9ca70d17090"},"schema_version":"1.0","source":{"id":"1908.02663","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.02663","created_at":"2026-07-05T00:03:17Z"},{"alias_kind":"arxiv_version","alias_value":"1908.02663v2","created_at":"2026-07-05T00:03:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.02663","created_at":"2026-07-05T00:03:17Z"},{"alias_kind":"pith_short_12","alias_value":"2BR2URIHQHIY","created_at":"2026-07-05T00:03:17Z"},{"alias_kind":"pith_short_16","alias_value":"2BR2URIHQHIYZJLZ","created_at":"2026-07-05T00:03:17Z"},{"alias_kind":"pith_short_8","alias_value":"2BR2URIH","created_at":"2026-07-05T00:03:17Z"}],"graph_snapshots":[{"event_id":"sha256:c000907305b18cd78a6f7d4140f1d9125f48ff51f45bdf72dbe86c544709f520","target":"graph","created_at":"2026-07-05T00:03:17Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.02663/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and speculated that it had a certain product formula involving the exponents of the group. We show that Molchanov's speculation is false in general but holds for all coincidental complex reflection groups when appropriately modified using exponents and co-exponents. These are the irreducible well-generated (i.e., duality) reflection groups with exponents forming an arithmetic progression and include many real r","authors_text":"Anne V. Shepler, Eric Sommers, Victor Reiner","cross_cats":["math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-07T14:44:09Z","title":"Invariant theory for coincidental complex reflection groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.02663","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":"sha256:1eebd69cf114b54cf89f72338a78e5d493054f95504906b9e4321629bcc86b5a","target":"record","created_at":"2026-07-05T00:03:17Z","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":"121a67bbba1172c43c6453a89eece9cf338630a1200999d9d61f590ce97c2745","cross_cats_sorted":["math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-08-07T14:44:09Z","title_canon_sha256":"8e0f9d4d0e050e9629d70c172653734608b5e35ce17cc78b2df6d9ca70d17090"},"schema_version":"1.0","source":{"id":"1908.02663","kind":"arxiv","version":2}},"canonical_sha256":"d063aa450781d18ca5796685cebe3cec1172e19eb8fadbd11ece67059274a3aa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d063aa450781d18ca5796685cebe3cec1172e19eb8fadbd11ece67059274a3aa","first_computed_at":"2026-07-05T00:03:17.021267Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:03:17.021267Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ivyKwZZUZsVLce4DLWfE1692WcO+nJs7ryDDq3ASQIoXp0JHSZmV424XzadZaELcXFaR58Pi9tqB0qtpf+44Dg==","signature_status":"signed_v1","signed_at":"2026-07-05T00:03:17.021683Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.02663","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1eebd69cf114b54cf89f72338a78e5d493054f95504906b9e4321629bcc86b5a","sha256:c000907305b18cd78a6f7d4140f1d9125f48ff51f45bdf72dbe86c544709f520"],"state_sha256":"0ce5d0dc0810135c83f4b9c49b477669a4cd702ca57a4e050a6c2072f5228c2f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xZG7pS8pnn/WZJQZjiv7x4iI2l5rxKNf/5NJgFJY7g2U8TofhHj6gi+SVi192vLQ6kxDktWvVzAoHPr+5ETtAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T13:37:45.844605Z","bundle_sha256":"393e767870b47d8a99024bf8e5e32c054fb1d50431799b5a9493176c88f41fc1"}}