{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:PGBYMZCHVQLRTTS3XRDQDU6J6Q","short_pith_number":"pith:PGBYMZCH","canonical_record":{"source":{"id":"1504.07318","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-28T00:51:55Z","cross_cats_sorted":[],"title_canon_sha256":"ebb1c7552934444c6237da02d1b2a443a92465cc7eff786b0fb9bc835e2d6ebc","abstract_canon_sha256":"b2f70358d36b14bd362c977219e203480723dd379ffaa2805e09cf87c794f850"},"schema_version":"1.0"},"canonical_sha256":"7983866447ac1719ce5bbc4701d3c9f438928c96a49ae0a44ab454ae6636afc1","source":{"kind":"arxiv","id":"1504.07318","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.07318","created_at":"2026-05-18T00:45:08Z"},{"alias_kind":"arxiv_version","alias_value":"1504.07318v2","created_at":"2026-05-18T00:45:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.07318","created_at":"2026-05-18T00:45:08Z"},{"alias_kind":"pith_short_12","alias_value":"PGBYMZCHVQLR","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PGBYMZCHVQLRTTS3","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PGBYMZCH","created_at":"2026-05-18T12:29:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:PGBYMZCHVQLRTTS3XRDQDU6J6Q","target":"record","payload":{"canonical_record":{"source":{"id":"1504.07318","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-28T00:51:55Z","cross_cats_sorted":[],"title_canon_sha256":"ebb1c7552934444c6237da02d1b2a443a92465cc7eff786b0fb9bc835e2d6ebc","abstract_canon_sha256":"b2f70358d36b14bd362c977219e203480723dd379ffaa2805e09cf87c794f850"},"schema_version":"1.0"},"canonical_sha256":"7983866447ac1719ce5bbc4701d3c9f438928c96a49ae0a44ab454ae6636afc1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:08.718972Z","signature_b64":"DTHKFxgeGmWYIelgK9Ibfi8XNDjtTuUMmk1LQOKR0GVqtO13Z/77bCfB0G+3FnPkbveTQ1ljFyJ6WP00ch2NAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7983866447ac1719ce5bbc4701d3c9f438928c96a49ae0a44ab454ae6636afc1","last_reissued_at":"2026-05-18T00:45:08.718328Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:08.718328Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.07318","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":"2026-05-18T00:45:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XhqofirK/4zriYW+YjxTpc7hefuVO+X51XdSnlpYxz14zVzJ3h4AH+CI2y8o0j9dzHmhFZHHJK7fOF3ziT7XCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T02:02:09.833925Z"},"content_sha256":"97dd29dd4ea4dc6b9d1d9233b3f686425c2767b2f15fd1c5d94b194a51155afe","schema_version":"1.0","event_id":"sha256:97dd29dd4ea4dc6b9d1d9233b3f686425c2767b2f15fd1c5d94b194a51155afe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:PGBYMZCHVQLRTTS3XRDQDU6J6Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalized Polarization Modules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hector Blandin","submitted_at":"2015-04-28T00:51:55Z","abstract_excerpt":"This work enrols the research line of M. Haiman on the Operator Theorem (the old operator conjecture). This theorem states that the smallest $\\mathfrak{S}_n$-module closed under taking partial derivatives and closed under the action of polarization operators that contains the Vandermonde determinant is the space of diagonal harmonics polynomials. We start generalizing the context of this theorem to the context of polynomials in $\\ell$ sets of $n$ variables $x_{ij}$ with $1\\leq i\\leq \\ell$ et $1\\leq j\\leq n$. Given a $\\mathfrak{S}_n$-stable family of homogeneous polynomials in the variables $x_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07318","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":"2026-05-18T00:45:08Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NVDbCmWskamkYUxTWog33gxDeePU0LFShXDzPZarEBej6EuZStf+IojaDJcA6HiCMfXiwHmhmyQkRato4edUAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T02:02:09.834643Z"},"content_sha256":"b239ea651f4a61f3aadb23299b6a028ef923a83c45756692615a0ff9a8dc4382","schema_version":"1.0","event_id":"sha256:b239ea651f4a61f3aadb23299b6a028ef923a83c45756692615a0ff9a8dc4382"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PGBYMZCHVQLRTTS3XRDQDU6J6Q/bundle.json","state_url":"https://pith.science/pith/PGBYMZCHVQLRTTS3XRDQDU6J6Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PGBYMZCHVQLRTTS3XRDQDU6J6Q/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":"2026-06-08T02:02:09Z","links":{"resolver":"https://pith.science/pith/PGBYMZCHVQLRTTS3XRDQDU6J6Q","bundle":"https://pith.science/pith/PGBYMZCHVQLRTTS3XRDQDU6J6Q/bundle.json","state":"https://pith.science/pith/PGBYMZCHVQLRTTS3XRDQDU6J6Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PGBYMZCHVQLRTTS3XRDQDU6J6Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:PGBYMZCHVQLRTTS3XRDQDU6J6Q","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":"b2f70358d36b14bd362c977219e203480723dd379ffaa2805e09cf87c794f850","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-28T00:51:55Z","title_canon_sha256":"ebb1c7552934444c6237da02d1b2a443a92465cc7eff786b0fb9bc835e2d6ebc"},"schema_version":"1.0","source":{"id":"1504.07318","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.07318","created_at":"2026-05-18T00:45:08Z"},{"alias_kind":"arxiv_version","alias_value":"1504.07318v2","created_at":"2026-05-18T00:45:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.07318","created_at":"2026-05-18T00:45:08Z"},{"alias_kind":"pith_short_12","alias_value":"PGBYMZCHVQLR","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_16","alias_value":"PGBYMZCHVQLRTTS3","created_at":"2026-05-18T12:29:37Z"},{"alias_kind":"pith_short_8","alias_value":"PGBYMZCH","created_at":"2026-05-18T12:29:37Z"}],"graph_snapshots":[{"event_id":"sha256:b239ea651f4a61f3aadb23299b6a028ef923a83c45756692615a0ff9a8dc4382","target":"graph","created_at":"2026-05-18T00:45:08Z","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 work enrols the research line of M. Haiman on the Operator Theorem (the old operator conjecture). This theorem states that the smallest $\\mathfrak{S}_n$-module closed under taking partial derivatives and closed under the action of polarization operators that contains the Vandermonde determinant is the space of diagonal harmonics polynomials. We start generalizing the context of this theorem to the context of polynomials in $\\ell$ sets of $n$ variables $x_{ij}$ with $1\\leq i\\leq \\ell$ et $1\\leq j\\leq n$. Given a $\\mathfrak{S}_n$-stable family of homogeneous polynomials in the variables $x_","authors_text":"Hector Blandin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-28T00:51:55Z","title":"Generalized Polarization Modules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07318","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":"sha256:97dd29dd4ea4dc6b9d1d9233b3f686425c2767b2f15fd1c5d94b194a51155afe","target":"record","created_at":"2026-05-18T00:45:08Z","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":"b2f70358d36b14bd362c977219e203480723dd379ffaa2805e09cf87c794f850","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-28T00:51:55Z","title_canon_sha256":"ebb1c7552934444c6237da02d1b2a443a92465cc7eff786b0fb9bc835e2d6ebc"},"schema_version":"1.0","source":{"id":"1504.07318","kind":"arxiv","version":2}},"canonical_sha256":"7983866447ac1719ce5bbc4701d3c9f438928c96a49ae0a44ab454ae6636afc1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7983866447ac1719ce5bbc4701d3c9f438928c96a49ae0a44ab454ae6636afc1","first_computed_at":"2026-05-18T00:45:08.718328Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:08.718328Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DTHKFxgeGmWYIelgK9Ibfi8XNDjtTuUMmk1LQOKR0GVqtO13Z/77bCfB0G+3FnPkbveTQ1ljFyJ6WP00ch2NAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:08.718972Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.07318","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:97dd29dd4ea4dc6b9d1d9233b3f686425c2767b2f15fd1c5d94b194a51155afe","sha256:b239ea651f4a61f3aadb23299b6a028ef923a83c45756692615a0ff9a8dc4382"],"state_sha256":"61c8eb367cf2c3344bfb4f78fb923a9fd5aead0b33469542f494a3ac6eca6cb7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eT/cUkmzW67awMQWMVkRHbzwUADUjCqFDTm7Prh+TDq7v9YRi/LC8xOEsJnT59Y5WjRlRVZsfM+NSC37QLneBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T02:02:09.838288Z","bundle_sha256":"cb20bc1e829f4e4139fe5a0cb283f5cd54db05e53455e61ddc452a6f3b98ba5a"}}