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Given a finite-dimensional module $M$ of the path algebra $kK(n)=\\mathcal{K}_n$, we consider its dimension vector $\\underline{dim} M=(\\dim_k M_1, \\dim_k M_2)$. Let $\\mathbf{F}=\\{(x,y)\\mid \\frac{2}{n}x\\leq y\\leq x\\}$, and let $(x,y)\\in\\mathbf{F}$. We construct a module $X(x,y)$ of $\\mathcal{K}_n$, and we prove it to be elementary. Suppose that $\\underline{dim} M=(x,y)$. We show that: if $M$ is an elementary modu"},"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":"2304.04182","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2023-04-09T07:51:10Z","cross_cats_sorted":[],"title_canon_sha256":"4e8718d0de4cd7cc2291ae46f09f1ab20da87ff4390db598221d06dbb58fb791","abstract_canon_sha256":"defa34eab0aa2d3791e0b9b09bd1aa49465cf5ba4a00346882f109c86e2fbc88"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:59:13.011952Z","signature_b64":"mhlwQ8mF4NG8OE/W4b776D1CIcIKG2l8JYMdJOhdwz/BQ+GJJ1q5KmpM5yKsYW88/pl5EpPtA2WF2Tbd977zCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47317761654776201137e2673845ce13e70615b8831000be167087336663127b","last_reissued_at":"2026-07-05T05:59:13.011488Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:59:13.011488Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dimension vectors of elementary modules of generalized Kronecker quivers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Jie Liu","submitted_at":"2023-04-09T07:51:10Z","abstract_excerpt":"Let $k$ be an algebraically closed field. The generalized or $n$-Kronecker quiver $K(n)$ is the quiver with two vertices, called a source and a sink, and $n$ arrows from source to sink. Given a finite-dimensional module $M$ of the path algebra $kK(n)=\\mathcal{K}_n$, we consider its dimension vector $\\underline{dim} M=(\\dim_k M_1, \\dim_k M_2)$. Let $\\mathbf{F}=\\{(x,y)\\mid \\frac{2}{n}x\\leq y\\leq x\\}$, and let $(x,y)\\in\\mathbf{F}$. We construct a module $X(x,y)$ of $\\mathcal{K}_n$, and we prove it to be elementary. Suppose that $\\underline{dim} M=(x,y)$. 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