{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:O4HYTDXY6YZ2D47MTZUXEU64MA","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":"6f9be3f56c8b8edef8a66c1f91981c835086bebd9e7eb2bbf96d23e95b0b08e6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-05-03T23:50:51Z","title_canon_sha256":"7f748e9645f9820793790c12185e97c504ed84b1ae4238695040d3b1b04b8b25"},"schema_version":"1.0","source":{"id":"1005.0422","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.0422","created_at":"2026-05-18T02:58:02Z"},{"alias_kind":"arxiv_version","alias_value":"1005.0422v2","created_at":"2026-05-18T02:58:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.0422","created_at":"2026-05-18T02:58:02Z"},{"alias_kind":"pith_short_12","alias_value":"O4HYTDXY6YZ2","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"O4HYTDXY6YZ2D47M","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"O4HYTDXY","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:bb3d3e70873aa514e1346005f290ddec83779876a823b9a34efa22eb8fc07572","target":"graph","created_at":"2026-05-18T02:58:02Z","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 $G$ be the universal Chevalley-Demazure group scheme corresponding to a reduced irreducible root system of rank $\\geq 2$, and let $R$ be a commutative ring. We analyze the linear representations $\\rho \\colon G(R)^+ \\to GL_n (K)$ over an algebraically closed field $K$ of the elementary subgroup $G(R)^+ \\subset G(R).$ Our main result is that under certain conditions, any such representation has a standard description, i.e. there exists a commutative finite-dimensional $K$-algebra $B$, a ring homomorphism $f \\colon R \\to B$ with Zariski-dense image, and a morphism of algebraic groups $\\sigma ","authors_text":"Igor A. Rapinchuk","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-05-03T23:50:51Z","title":"On linear representations of Chevalley groups over commutative rings"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.0422","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":"sha256:27b4d40e481fce69d778af3d07e86af5109c09f9a33a114993811cf9e60da878","target":"record","created_at":"2026-05-18T02:58:02Z","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":"6f9be3f56c8b8edef8a66c1f91981c835086bebd9e7eb2bbf96d23e95b0b08e6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-05-03T23:50:51Z","title_canon_sha256":"7f748e9645f9820793790c12185e97c504ed84b1ae4238695040d3b1b04b8b25"},"schema_version":"1.0","source":{"id":"1005.0422","kind":"arxiv","version":2}},"canonical_sha256":"770f898ef8f633a1f3ec9e697253dc602ca4abafb73594d7096852cff5a1ae91","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"770f898ef8f633a1f3ec9e697253dc602ca4abafb73594d7096852cff5a1ae91","first_computed_at":"2026-05-18T02:58:02.171976Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:02.171976Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0SFxkw4oXgLcTsdTyF7atM95/sSQ7ISs/aVYwyQXwTPwhx9G6wf//Aaexb9aCtn05l6oWPWZY/IreHye+lzNDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:02.172550Z","signed_message":"canonical_sha256_bytes"},"source_id":"1005.0422","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:27b4d40e481fce69d778af3d07e86af5109c09f9a33a114993811cf9e60da878","sha256:bb3d3e70873aa514e1346005f290ddec83779876a823b9a34efa22eb8fc07572"],"state_sha256":"24bb23fdfad6111bee57345c1bf5543d64f09dd68ff0d68d24ef309416b214b3"}