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We study in this note the reducibility of $C_r(\\N_n)$ for various values of $n$ and $r$. In particular it will be shown that the reducibility of $C_r(\\mathfrak{gl}_n)$, the variety of commuting $r$-tuples of $n$ by $n$ matrices, implies that of $C_r(\\N_n)$ under certain condition. Then we prove that $C_r(\\N_n)$ is reduc"},"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":"1308.2420","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-08-11T18:57:56Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"d9b615cf1a00622dbf019d25613dd7db2916acc8c4d988c03eff43f81ca9bafc","abstract_canon_sha256":"de895ae1ffb10f070bc80800f13d5a6a36586ea7bb517b8b96685d66b97ffcb9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:16:07.470201Z","signature_b64":"hkl/IC/Z29HblaMJzVQphg08QboowhZOPe+DkKeqmVn6X0n+7H7TsINxZtg9KAGCNsEXJU8j0t5bKTmDNgM+Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"901fbd57e785945a6c7dc6aaaeba06496600479ace0fdcafa276ae4aa27c5eac","last_reissued_at":"2026-05-18T03:16:07.469445Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:16:07.469445Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reducibility of nilpotent commuting varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.RT","authors_text":"Nham V. Ngo, Robert M. Guralnick","submitted_at":"2013-08-11T18:57:56Z","abstract_excerpt":"Let $\\N_n$ be the set of nilpotent $n$ by $n$ matrices over an algebraically closed field $k$. For each $r\\ge 2$, let $C_r(\\N_n)$ be the variety consisting of all pairwise commuting $r$-tuples of nilpotent matrices. It is well-kown that $C_2(\\N_n)$ is irreducible for every $n$. We study in this note the reducibility of $C_r(\\N_n)$ for various values of $n$ and $r$. In particular it will be shown that the reducibility of $C_r(\\mathfrak{gl}_n)$, the variety of commuting $r$-tuples of $n$ by $n$ matrices, implies that of $C_r(\\N_n)$ under certain condition. 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