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For each nilpotent orbit G.e of \\g, we list the Jordan blocks of the action of e on the minimal induced module V_min of \\g. We also establish when the centralisers G_v of vectors v\\in V_min and stabilisers \\Stab_G<v> of 1-spaces <v>\\subset V_min are smooth; that is, when \\dim G_v=\\dim\\g_v or \\dim \\Stab_G<v>=\\dim\\Stab_\\g<v>."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1508.02918","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2015-08-12T13:47:49Z","cross_cats_sorted":["math.GR","math.RA"],"title_canon_sha256":"b49a937c4dd7fe1a407c55715cfd9b6fbad98b72a9ff3ac2cba5af9c2d3285d8","abstract_canon_sha256":"1631e1b393fd9ebf3e6997e2d9b48277f6c54141157f485cbbe1c32e73dc619f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:10:51.702831Z","signature_b64":"T4nbXPfdQlRVWO0XCFiGdB8vUvBn1No1XZh5wbp1GEBgdWzr7QT3hSjZ2D+MZVRv95KuAXyGvEUzHR95OVRuDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0649f00899cf26e27820081e99828d591afa9f90d044666d8ff883837251ab5","last_reissued_at":"2026-07-05T08:10:51.702407Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:10:51.702407Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the minimal modules for exceptional Lie algebras: Jordan blocks and stabilisers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.RA"],"primary_cat":"math.RT","authors_text":"David I. Stewart","submitted_at":"2015-08-12T13:47:49Z","abstract_excerpt":"Let G be a simple simple-connected exceptional algebraic group of type G_2, F_4, E_6 or E_7 over an algebraically closed field k of characteristic p>0 with \\g=Lie(G). For each nilpotent orbit G.e of \\g, we list the Jordan blocks of the action of e on the minimal induced module V_min of \\g. We also establish when the centralisers G_v of vectors v\\in V_min and stabilisers \\Stab_G<v> of 1-spaces <v>\\subset V_min are smooth; that is, when \\dim G_v=\\dim\\g_v or \\dim \\Stab_G<v>=\\dim\\Stab_\\g<v>."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02918","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1508.02918/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1508.02918","created_at":"2026-07-05T08:10:51.702462+00:00"},{"alias_kind":"arxiv_version","alias_value":"1508.02918v6","created_at":"2026-07-05T08:10:51.702462+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02918","created_at":"2026-07-05T08:10:51.702462+00:00"},{"alias_kind":"pith_short_12","alias_value":"6BSJ6AEJTTZG","created_at":"2026-07-05T08:10:51.702462+00:00"},{"alias_kind":"pith_short_16","alias_value":"6BSJ6AEJTTZG4J4C","created_at":"2026-07-05T08:10:51.702462+00:00"},{"alias_kind":"pith_short_8","alias_value":"6BSJ6AEJ","created_at":"2026-07-05T08:10:51.702462+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.10585","citing_title":"Small Lie algebras in the Verlinde category","ref_index":29,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6BSJ6AEJTTZG4J4CACA6TGBI2W","json":"https://pith.science/pith/6BSJ6AEJTTZG4J4CACA6TGBI2W.json","graph_json":"https://pith.science/api/pith-number/6BSJ6AEJTTZG4J4CACA6TGBI2W/graph.json","events_json":"https://pith.science/api/pith-number/6BSJ6AEJTTZG4J4CACA6TGBI2W/events.json","paper":"https://pith.science/paper/6BSJ6AEJ"},"agent_actions":{"view_html":"https://pith.science/pith/6BSJ6AEJTTZG4J4CACA6TGBI2W","download_json":"https://pith.science/pith/6BSJ6AEJTTZG4J4CACA6TGBI2W.json","view_paper":"https://pith.science/paper/6BSJ6AEJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1508.02918&json=true","fetch_graph":"https://pith.science/api/pith-number/6BSJ6AEJTTZG4J4CACA6TGBI2W/graph.json","fetch_events":"https://pith.science/api/pith-number/6BSJ6AEJTTZG4J4CACA6TGBI2W/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6BSJ6AEJTTZG4J4CACA6TGBI2W/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6BSJ6AEJTTZG4J4CACA6TGBI2W/action/storage_attestation","attest_author":"https://pith.science/pith/6BSJ6AEJTTZG4J4CACA6TGBI2W/action/author_attestation","sign_citation":"https://pith.science/pith/6BSJ6AEJTTZG4J4CACA6TGBI2W/action/citation_signature","submit_replication":"https://pith.science/pith/6BSJ6AEJTTZG4J4CACA6TGBI2W/action/replication_record"}},"created_at":"2026-07-05T08:10:51.702462+00:00","updated_at":"2026-07-05T08:10:51.702462+00:00"}