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We prove that, if F is a finite field, and Char F is invertible in K, then the K-linear abelian category Rep(F;K) is equivalent to the product, over all k=0,1,2, ..., of the categories of K[GL(k,F)]-modules.\n  As a consequence, if K is also a field, then small projectives are also injective in Rep(F;K), and will have finite length. Even more is true if Char K = 0: the category Rep(F;K) will be semisimple.\n  In a last section, we briefly discuss \"q=1"},"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":"1405.0318","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2014-05-01T22:31:32Z","cross_cats_sorted":[],"title_canon_sha256":"fe28cbcf1b5ba9c5879dd39b33a45f86a7e57b674c9cc6359f937f1860c0d460","abstract_canon_sha256":"ac2f7c5dbfb3620639ab8e85b9c8b2321ffe3d28751a907dfa49975b6c740a3d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:52:25.035417Z","signature_b64":"U8cEuq5mOJdgNGFuAD7mF3A9e7WQ40YcC/FD30Jc6Uv6kRqogSK73yktvydNLHCkL64z30yQmm76RHEmwGxWCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"51a639172094d1ed0587ad71b198a4e59d1f5d262a0da3ed7e5939a30ac007cf","last_reissued_at":"2026-05-18T02:52:25.034767Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:52:25.034767Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generic representation theory of finite fields in nondescribing characteristic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Nicholas J. Kuhn","submitted_at":"2014-05-01T22:31:32Z","abstract_excerpt":"Let Rep(F;K) denote the category of functors from finite dimensional F-vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and Char F is invertible in K, then the K-linear abelian category Rep(F;K) is equivalent to the product, over all k=0,1,2, ..., of the categories of K[GL(k,F)]-modules.\n  As a consequence, if K is also a field, then small projectives are also injective in Rep(F;K), and will have finite length. Even more is true if Char K = 0: the category Rep(F;K) will be semisimple.\n  In a last section, we briefly discuss \"q=1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.0318","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":"1405.0318","created_at":"2026-05-18T02:52:25.034884+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.0318v2","created_at":"2026-05-18T02:52:25.034884+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.0318","created_at":"2026-05-18T02:52:25.034884+00:00"},{"alias_kind":"pith_short_12","alias_value":"KGTDSFZASTI6","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_16","alias_value":"KGTDSFZASTI62BMH","created_at":"2026-05-18T12:28:35.611951+00:00"},{"alias_kind":"pith_short_8","alias_value":"KGTDSFZA","created_at":"2026-05-18T12:28:35.611951+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KGTDSFZASTI62BMHVVY3DGFE4W","json":"https://pith.science/pith/KGTDSFZASTI62BMHVVY3DGFE4W.json","graph_json":"https://pith.science/api/pith-number/KGTDSFZASTI62BMHVVY3DGFE4W/graph.json","events_json":"https://pith.science/api/pith-number/KGTDSFZASTI62BMHVVY3DGFE4W/events.json","paper":"https://pith.science/paper/KGTDSFZA"},"agent_actions":{"view_html":"https://pith.science/pith/KGTDSFZASTI62BMHVVY3DGFE4W","download_json":"https://pith.science/pith/KGTDSFZASTI62BMHVVY3DGFE4W.json","view_paper":"https://pith.science/paper/KGTDSFZA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.0318&json=true","fetch_graph":"https://pith.science/api/pith-number/KGTDSFZASTI62BMHVVY3DGFE4W/graph.json","fetch_events":"https://pith.science/api/pith-number/KGTDSFZASTI62BMHVVY3DGFE4W/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KGTDSFZASTI62BMHVVY3DGFE4W/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KGTDSFZASTI62BMHVVY3DGFE4W/action/storage_attestation","attest_author":"https://pith.science/pith/KGTDSFZASTI62BMHVVY3DGFE4W/action/author_attestation","sign_citation":"https://pith.science/pith/KGTDSFZASTI62BMHVVY3DGFE4W/action/citation_signature","submit_replication":"https://pith.science/pith/KGTDSFZASTI62BMHVVY3DGFE4W/action/replication_record"}},"created_at":"2026-05-18T02:52:25.034884+00:00","updated_at":"2026-05-18T02:52:25.034884+00:00"}