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We prove that the density of non-cyclic matrices in ${\\rm M}(V)_U$ is at least $q^{-2}\\left(1+c_1q^{-1}\\right)$, and at most $q^{-2}\\left(1+c_2q^{-1}\\right)$, where $c_1$ and $c_2$ are constants independent of $n,r$, and $q$. The constants $c_1=-\\frac43$ and $c_2=\\frac{35}3$ suffice."},"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":"1405.6609","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2014-05-22T05:01:27Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"0253e9dfda57424ac92ad655843b9f2c14910d834c8c38d9d236efffe138da65","abstract_canon_sha256":"6cc13e89369ec07d3bc4e5ae39caaa6137f3b2d1fcc03d95dc4ebe47a89b5652"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:46.586479Z","signature_b64":"6q7XBIyoRXMY5iJExFtkpUizrwkc28+AjKphbwohQOnxq57+CfKP1pdISNYDE6SXuZKVusrFQjRclXlsMzV6Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b5f80995adfe62f7914cdf89d33a3cb6b5122d6c2d746624e0bd51dcec0ef440","last_reissued_at":"2026-05-18T01:05:46.585826Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:46.585826Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proportion of cyclic matrices in maximal reducible matrix algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.RA","authors_text":"Cheryl E. Praeger, Michael Giudici, Scott Brown, S.P. Glasby","submitted_at":"2014-05-22T05:01:27Z","abstract_excerpt":"Let ${\\rm M}(V)={\\rm M}(n,\\mathbb{F}_q)$ denote the algebra of $n\\times n$ matrices over $\\mathbb{F}_q$, and let ${\\rm M}(V)_U$ denote the (maximal reducible) subalgebra that normalizes a given $r$-dimensional subspace $U$ of $V=\\mathbb{F}_q^n$ where $0<r<n$. We prove that the density of non-cyclic matrices in ${\\rm M}(V)_U$ is at least $q^{-2}\\left(1+c_1q^{-1}\\right)$, and at most $q^{-2}\\left(1+c_2q^{-1}\\right)$, where $c_1$ and $c_2$ are constants independent of $n,r$, and $q$. The constants $c_1=-\\frac43$ and $c_2=\\frac{35}3$ suffice."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.6609","kind":"arxiv","version":1},"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":"1405.6609","created_at":"2026-05-18T01:05:46.585925+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.6609v1","created_at":"2026-05-18T01:05:46.585925+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.6609","created_at":"2026-05-18T01:05:46.585925+00:00"},{"alias_kind":"pith_short_12","alias_value":"WX4ATFNN7ZRP","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_16","alias_value":"WX4ATFNN7ZRPPEKM","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_8","alias_value":"WX4ATFNN","created_at":"2026-05-18T12:28:54.890064+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/WX4ATFNN7ZRPPEKM36E5GOR4W2","json":"https://pith.science/pith/WX4ATFNN7ZRPPEKM36E5GOR4W2.json","graph_json":"https://pith.science/api/pith-number/WX4ATFNN7ZRPPEKM36E5GOR4W2/graph.json","events_json":"https://pith.science/api/pith-number/WX4ATFNN7ZRPPEKM36E5GOR4W2/events.json","paper":"https://pith.science/paper/WX4ATFNN"},"agent_actions":{"view_html":"https://pith.science/pith/WX4ATFNN7ZRPPEKM36E5GOR4W2","download_json":"https://pith.science/pith/WX4ATFNN7ZRPPEKM36E5GOR4W2.json","view_paper":"https://pith.science/paper/WX4ATFNN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.6609&json=true","fetch_graph":"https://pith.science/api/pith-number/WX4ATFNN7ZRPPEKM36E5GOR4W2/graph.json","fetch_events":"https://pith.science/api/pith-number/WX4ATFNN7ZRPPEKM36E5GOR4W2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WX4ATFNN7ZRPPEKM36E5GOR4W2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WX4ATFNN7ZRPPEKM36E5GOR4W2/action/storage_attestation","attest_author":"https://pith.science/pith/WX4ATFNN7ZRPPEKM36E5GOR4W2/action/author_attestation","sign_citation":"https://pith.science/pith/WX4ATFNN7ZRPPEKM36E5GOR4W2/action/citation_signature","submit_replication":"https://pith.science/pith/WX4ATFNN7ZRPPEKM36E5GOR4W2/action/replication_record"}},"created_at":"2026-05-18T01:05:46.585925+00:00","updated_at":"2026-05-18T01:05:46.585925+00:00"}