{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:UXXLHIOWOCTBVF7Y2LNUA4TXXN","short_pith_number":"pith:UXXLHIOW","canonical_record":{"source":{"id":"0709.2202","kind":"arxiv","version":6},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2007-09-14T17:38:07Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"5f2c7cc41f62f24d8fc6bf4e18015cfa94e421ac6e9e2d2e8a350854f3667a99","abstract_canon_sha256":"cfa6cdc0921f991541599f883386c2057fe836b1b8128b42292cb3f4d3969469"},"schema_version":"1.0"},"canonical_sha256":"a5eeb3a1d670a61a97f8d2db407277bb650a5dea7784d23c5b3166b74831bcff","source":{"kind":"arxiv","id":"0709.2202","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0709.2202","created_at":"2026-05-18T04:09:14Z"},{"alias_kind":"arxiv_version","alias_value":"0709.2202v6","created_at":"2026-05-18T04:09:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0709.2202","created_at":"2026-05-18T04:09:14Z"},{"alias_kind":"pith_short_12","alias_value":"UXXLHIOWOCTB","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"UXXLHIOWOCTBVF7Y","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"UXXLHIOW","created_at":"2026-05-18T12:25:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:UXXLHIOWOCTBVF7Y2LNUA4TXXN","target":"record","payload":{"canonical_record":{"source":{"id":"0709.2202","kind":"arxiv","version":6},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2007-09-14T17:38:07Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"5f2c7cc41f62f24d8fc6bf4e18015cfa94e421ac6e9e2d2e8a350854f3667a99","abstract_canon_sha256":"cfa6cdc0921f991541599f883386c2057fe836b1b8128b42292cb3f4d3969469"},"schema_version":"1.0"},"canonical_sha256":"a5eeb3a1d670a61a97f8d2db407277bb650a5dea7784d23c5b3166b74831bcff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:14.542147Z","signature_b64":"2cPftl82S0E8gLJWyNI9euzSjiOKoxQsap+dIby919G8zLInRk5EGppaAR1bosGZgD0L4KGrREpXMxRdETHUAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a5eeb3a1d670a61a97f8d2db407277bb650a5dea7784d23c5b3166b74831bcff","last_reissued_at":"2026-05-18T04:09:14.541722Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:14.541722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0709.2202","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:09:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FQv+vGLk+S+dp1WaQK7nu08ptpgpFkd8YGSJx9AMbpUXmfYhbvgW6KUTOiZKwFzrzlh1ktJ42WG+EhrvnNLeDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T16:34:45.977379Z"},"content_sha256":"3ff9851b3ce8ec5f45a4868107ab972cec5e94d006ac890bf788fa829bb928da","schema_version":"1.0","event_id":"sha256:3ff9851b3ce8ec5f45a4868107ab972cec5e94d006ac890bf788fa829bb928da"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:UXXLHIOWOCTBVF7Y2LNUA4TXXN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Linear maps preserving fibers","license":"","headline":"","cross_cats":["math.RT"],"primary_cat":"math.GR","authors_text":"Gerald W. Schwarz","submitted_at":"2007-09-14T17:38:07Z","abstract_excerpt":"Let $G\\subset\\GL(V)$ be a complex reductive group where $\\dim V<\\infty$, and let $\\pi\\colon V\\to\\quot VG$ be the categorical quotient. Let $\\NN:=\\pi\\inv\\pi(0)$ be the null cone of $V$, let $H_0$ be the subgroup of $\\GL(V)$ which preserves the ideal $\\I$ of $\\NN$ and let $H$ be a Levi subgroup of $H_0$ containing $G$. We determine the identity component of $H$. In many cases we show that $H=H_0$. For adjoint representations we have $H=H_0$ and we determine $H$ completely. We also investigate the subgroup $G_F$ of $\\GL(V)$ preserving a fiber $F$ of $\\pi$ when $V$ is an irreducible cofree $G$-mod"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.2202","kind":"arxiv","version":6},"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":"2026-05-18T04:09:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JCCqXNLn9YauqiHuo9yzaDwuq/XE6V6l0IsBhNrXq34efjPv/1iDxFQOjOuy2GTMBFpwk/poZWqJ9OH6775qDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T16:34:45.977730Z"},"content_sha256":"322b67e5205f86df07456e352c699751814fe7cb0eab9c77d09e42f1cf749263","schema_version":"1.0","event_id":"sha256:322b67e5205f86df07456e352c699751814fe7cb0eab9c77d09e42f1cf749263"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UXXLHIOWOCTBVF7Y2LNUA4TXXN/bundle.json","state_url":"https://pith.science/pith/UXXLHIOWOCTBVF7Y2LNUA4TXXN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UXXLHIOWOCTBVF7Y2LNUA4TXXN/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":"2026-06-28T16:34:45Z","links":{"resolver":"https://pith.science/pith/UXXLHIOWOCTBVF7Y2LNUA4TXXN","bundle":"https://pith.science/pith/UXXLHIOWOCTBVF7Y2LNUA4TXXN/bundle.json","state":"https://pith.science/pith/UXXLHIOWOCTBVF7Y2LNUA4TXXN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UXXLHIOWOCTBVF7Y2LNUA4TXXN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:UXXLHIOWOCTBVF7Y2LNUA4TXXN","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":"cfa6cdc0921f991541599f883386c2057fe836b1b8128b42292cb3f4d3969469","cross_cats_sorted":["math.RT"],"license":"","primary_cat":"math.GR","submitted_at":"2007-09-14T17:38:07Z","title_canon_sha256":"5f2c7cc41f62f24d8fc6bf4e18015cfa94e421ac6e9e2d2e8a350854f3667a99"},"schema_version":"1.0","source":{"id":"0709.2202","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0709.2202","created_at":"2026-05-18T04:09:14Z"},{"alias_kind":"arxiv_version","alias_value":"0709.2202v6","created_at":"2026-05-18T04:09:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0709.2202","created_at":"2026-05-18T04:09:14Z"},{"alias_kind":"pith_short_12","alias_value":"UXXLHIOWOCTB","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_16","alias_value":"UXXLHIOWOCTBVF7Y","created_at":"2026-05-18T12:25:56Z"},{"alias_kind":"pith_short_8","alias_value":"UXXLHIOW","created_at":"2026-05-18T12:25:56Z"}],"graph_snapshots":[{"event_id":"sha256:322b67e5205f86df07456e352c699751814fe7cb0eab9c77d09e42f1cf749263","target":"graph","created_at":"2026-05-18T04:09:14Z","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 $G\\subset\\GL(V)$ be a complex reductive group where $\\dim V<\\infty$, and let $\\pi\\colon V\\to\\quot VG$ be the categorical quotient. Let $\\NN:=\\pi\\inv\\pi(0)$ be the null cone of $V$, let $H_0$ be the subgroup of $\\GL(V)$ which preserves the ideal $\\I$ of $\\NN$ and let $H$ be a Levi subgroup of $H_0$ containing $G$. We determine the identity component of $H$. In many cases we show that $H=H_0$. For adjoint representations we have $H=H_0$ and we determine $H$ completely. We also investigate the subgroup $G_F$ of $\\GL(V)$ preserving a fiber $F$ of $\\pi$ when $V$ is an irreducible cofree $G$-mod","authors_text":"Gerald W. Schwarz","cross_cats":["math.RT"],"headline":"","license":"","primary_cat":"math.GR","submitted_at":"2007-09-14T17:38:07Z","title":"Linear maps preserving fibers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.2202","kind":"arxiv","version":6},"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":"sha256:3ff9851b3ce8ec5f45a4868107ab972cec5e94d006ac890bf788fa829bb928da","target":"record","created_at":"2026-05-18T04:09:14Z","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":"cfa6cdc0921f991541599f883386c2057fe836b1b8128b42292cb3f4d3969469","cross_cats_sorted":["math.RT"],"license":"","primary_cat":"math.GR","submitted_at":"2007-09-14T17:38:07Z","title_canon_sha256":"5f2c7cc41f62f24d8fc6bf4e18015cfa94e421ac6e9e2d2e8a350854f3667a99"},"schema_version":"1.0","source":{"id":"0709.2202","kind":"arxiv","version":6}},"canonical_sha256":"a5eeb3a1d670a61a97f8d2db407277bb650a5dea7784d23c5b3166b74831bcff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a5eeb3a1d670a61a97f8d2db407277bb650a5dea7784d23c5b3166b74831bcff","first_computed_at":"2026-05-18T04:09:14.541722Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:09:14.541722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2cPftl82S0E8gLJWyNI9euzSjiOKoxQsap+dIby919G8zLInRk5EGppaAR1bosGZgD0L4KGrREpXMxRdETHUAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:09:14.542147Z","signed_message":"canonical_sha256_bytes"},"source_id":"0709.2202","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3ff9851b3ce8ec5f45a4868107ab972cec5e94d006ac890bf788fa829bb928da","sha256:322b67e5205f86df07456e352c699751814fe7cb0eab9c77d09e42f1cf749263"],"state_sha256":"0c3481a7287b5d9c4ffae8e2f4c03c3da7b37d7440b29130546dc8a1756a9772"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g7pho7M7ZZZk/qwpGqZ4z0qBO7KDHfsVgHtxIw+nMYALQlnHsX8cSvmzzaSHwydfvMJGb+/pFP/sZnlSX8lKCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T16:34:45.979745Z","bundle_sha256":"c32716b43252efd11d357ca699c16cd511e17466c93b60a3f529f79e245d76d1"}}