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We study in this paper the reducibility of $N(d,n)$ for various values of $d$ and $n$. In particular, we prove that $N(d,n)$ is reducible for all $d,n\\ge 4$. In the case $d=3$, we show that it is irreducible for $n\\le 6$."},"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":"1308.4438","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-08-20T21:29:12Z","cross_cats_sorted":[],"title_canon_sha256":"3d395b5dc603e59ab974158ad7344ffd7cea8a8787d567865f984387eed5705c","abstract_canon_sha256":"d7e8954198286dbc25c1b020b371a02a03d516b6a64db951ecca1cceeb8b5b29"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:55:27.552344Z","signature_b64":"pQM1crI2Cow5UGW2egaVPElSSVkmp7MgB+29sgYPeqLBCI9fBgRPac4JKzKRQekdpWSlM+k7sJFr2Bsh2oS5DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"834cbe94596d9f787f89ac16a0586de4850e490948d4baf7ede1d523003657a9","last_reissued_at":"2026-05-18T02:55:27.551736Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:55:27.551736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On varieties of commuting nilpotent matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Klemen \\v{S}ivic, Nham V. Ngo","submitted_at":"2013-08-20T21:29:12Z","abstract_excerpt":"Let $N(d,n)$ be the variety of all $d$-tuples of commuting nilpotent $n\\times n$ matrices. It is well-known that $N(d,n)$ is irreducible if $d=2$, if $n\\le 3$ or if $d=3$ and $n=4$. On the other hand $N(3,n)$ is known to be reducible for $n\\ge 13$. We study in this paper the reducibility of $N(d,n)$ for various values of $d$ and $n$. In particular, we prove that $N(d,n)$ is reducible for all $d,n\\ge 4$. In the case $d=3$, we show that it is irreducible for $n\\le 6$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4438","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":"1308.4438","created_at":"2026-05-18T02:55:27.551830+00:00"},{"alias_kind":"arxiv_version","alias_value":"1308.4438v2","created_at":"2026-05-18T02:55:27.551830+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1308.4438","created_at":"2026-05-18T02:55:27.551830+00:00"},{"alias_kind":"pith_short_12","alias_value":"QNGL5FCZNWPX","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_16","alias_value":"QNGL5FCZNWPXQ74J","created_at":"2026-05-18T12:27:57.521954+00:00"},{"alias_kind":"pith_short_8","alias_value":"QNGL5FCZ","created_at":"2026-05-18T12:27:57.521954+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/QNGL5FCZNWPXQ74JVQLKAWDN4S","json":"https://pith.science/pith/QNGL5FCZNWPXQ74JVQLKAWDN4S.json","graph_json":"https://pith.science/api/pith-number/QNGL5FCZNWPXQ74JVQLKAWDN4S/graph.json","events_json":"https://pith.science/api/pith-number/QNGL5FCZNWPXQ74JVQLKAWDN4S/events.json","paper":"https://pith.science/paper/QNGL5FCZ"},"agent_actions":{"view_html":"https://pith.science/pith/QNGL5FCZNWPXQ74JVQLKAWDN4S","download_json":"https://pith.science/pith/QNGL5FCZNWPXQ74JVQLKAWDN4S.json","view_paper":"https://pith.science/paper/QNGL5FCZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1308.4438&json=true","fetch_graph":"https://pith.science/api/pith-number/QNGL5FCZNWPXQ74JVQLKAWDN4S/graph.json","fetch_events":"https://pith.science/api/pith-number/QNGL5FCZNWPXQ74JVQLKAWDN4S/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QNGL5FCZNWPXQ74JVQLKAWDN4S/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QNGL5FCZNWPXQ74JVQLKAWDN4S/action/storage_attestation","attest_author":"https://pith.science/pith/QNGL5FCZNWPXQ74JVQLKAWDN4S/action/author_attestation","sign_citation":"https://pith.science/pith/QNGL5FCZNWPXQ74JVQLKAWDN4S/action/citation_signature","submit_replication":"https://pith.science/pith/QNGL5FCZNWPXQ74JVQLKAWDN4S/action/replication_record"}},"created_at":"2026-05-18T02:55:27.551830+00:00","updated_at":"2026-05-18T02:55:27.551830+00:00"}