{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:GI3Y76FZ7BYJKZYE6PGJ7RR5Y2","short_pith_number":"pith:GI3Y76FZ","schema_version":"1.0","canonical_sha256":"32378ff8b9f870956704f3cc9fc63dc68f036110c7c9b9c8f3f13a1445d5c675","source":{"kind":"arxiv","id":"1010.0811","version":2},"attestation_state":"computed","paper":{"title":"Algebraic zip data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.RT","authors_text":"Paul Ziegler, Richard Pink, Torsten Wedhorn","submitted_at":"2010-10-05T09:25:14Z","abstract_excerpt":"An algebraic zip datum is a tuple $\\CZ := (G,P,Q,\\phi)$ consisting of a reductive group $G$ together with parabolic subgroups $P$ and $Q$ and an isogeny $\\phi\\colon P/R_uP\\to Q/R_uQ$. We study the action of the group $E := \\{(p,q)\\in P{\\times}Q | \\phi(\\pi_{P}(p)) =\\pi_Q(q)\\}$ on $G$ given by $((p,q),g)\\mapsto pgq^{-1}$. We define certain smooth $E$-invariant subvarieties of $G$, show that they define a stratification of $G$. We determine their dimensions and their closures and give a description of the stabilizers of the $E$-action on $G$. We also generalize all results to non-connected groups"},"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":"1010.0811","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2010-10-05T09:25:14Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"a933a738dc626760f715724920aa473842051b6aae2d26966abbc7849267a20c","abstract_canon_sha256":"16467a665c564752f063ed9667db76cd9eb5553a60bd3cef6e980bee1ec6eb65"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:32.969645Z","signature_b64":"lUt3HT2eCGTYK0uy8P2wqAiFayx32Da1yjLnZaamrPZZGh8U/srtthvGYkfhyRgLhETto43R0jW5CeIYR61MAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"32378ff8b9f870956704f3cc9fc63dc68f036110c7c9b9c8f3f13a1445d5c675","last_reissued_at":"2026-05-18T04:26:32.969163Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:32.969163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Algebraic zip data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.RT","authors_text":"Paul Ziegler, Richard Pink, Torsten Wedhorn","submitted_at":"2010-10-05T09:25:14Z","abstract_excerpt":"An algebraic zip datum is a tuple $\\CZ := (G,P,Q,\\phi)$ consisting of a reductive group $G$ together with parabolic subgroups $P$ and $Q$ and an isogeny $\\phi\\colon P/R_uP\\to Q/R_uQ$. We study the action of the group $E := \\{(p,q)\\in P{\\times}Q | \\phi(\\pi_{P}(p)) =\\pi_Q(q)\\}$ on $G$ given by $((p,q),g)\\mapsto pgq^{-1}$. We define certain smooth $E$-invariant subvarieties of $G$, show that they define a stratification of $G$. We determine their dimensions and their closures and give a description of the stabilizers of the $E$-action on $G$. We also generalize all results to non-connected groups"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.0811","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1010.0811","created_at":"2026-05-18T04:26:32.969233+00:00"},{"alias_kind":"arxiv_version","alias_value":"1010.0811v2","created_at":"2026-05-18T04:26:32.969233+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.0811","created_at":"2026-05-18T04:26:32.969233+00:00"},{"alias_kind":"pith_short_12","alias_value":"GI3Y76FZ7BYJ","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_16","alias_value":"GI3Y76FZ7BYJKZYE","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_8","alias_value":"GI3Y76FZ","created_at":"2026-05-18T12:26:07.630475+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2509.19878","citing_title":"Intersections of the Ekedahl-Oort and Newton Strata of $\\mathcal{A}_{5}$","ref_index":13,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2","json":"https://pith.science/pith/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2.json","graph_json":"https://pith.science/api/pith-number/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2/graph.json","events_json":"https://pith.science/api/pith-number/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2/events.json","paper":"https://pith.science/paper/GI3Y76FZ"},"agent_actions":{"view_html":"https://pith.science/pith/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2","download_json":"https://pith.science/pith/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2.json","view_paper":"https://pith.science/paper/GI3Y76FZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1010.0811&json=true","fetch_graph":"https://pith.science/api/pith-number/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2/graph.json","fetch_events":"https://pith.science/api/pith-number/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2/action/storage_attestation","attest_author":"https://pith.science/pith/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2/action/author_attestation","sign_citation":"https://pith.science/pith/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2/action/citation_signature","submit_replication":"https://pith.science/pith/GI3Y76FZ7BYJKZYE6PGJ7RR5Y2/action/replication_record"}},"created_at":"2026-05-18T04:26:32.969233+00:00","updated_at":"2026-05-18T04:26:32.969233+00:00"}