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We show that $\\cW(\\cD)\\geq \\frac{b(\\cD)+1}{2}$. In particular, we get $\\{\\cW(\\cD)| \\cD \\text{ is a regular component} \\}=\\mathbb{N}$."},"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":"2606.06964","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RT","submitted_at":"2026-06-05T06:42:38Z","cross_cats_sorted":[],"title_canon_sha256":"323ad243419fbb06c2ef99e9e33527b5b5551641e85f276b46dcf9fd30fd2834","abstract_canon_sha256":"d0c0015b9868f464d05fc8825eeae0ee9e518a4aec69ebd7001a5630cdc7ecb9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-08T01:04:38.471185Z","signature_b64":"kLqbPSzbvkdUoqs+ArpgE4xCfj9dJho0w8R7jqvFOihUgMHcXFN2i108Jh58IxGRzv7pNA0HyWvAx2DqSTv0Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f08098b5f70d79c14aa437a3241c8101965b0374232f6462330c15228881586","last_reissued_at":"2026-06-08T01:04:38.470436Z","signature_status":"signed_v1","first_computed_at":"2026-06-08T01:04:38.470436Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Widths of regular components for n-regular tree $T(n)$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Jie Liu","submitted_at":"2026-06-05T06:42:38Z","abstract_excerpt":"Let $(T(n),\\Omega)$ be the covering of the generalized Kronecker quiver $K(n)$, where $\\Omega$ is a bipartite orientation. Given a regular Auslander--Reiten component $\\cD$ of $\\modd(T(n),\\Omega)$, we introduce two invariants: the width $\\cW(\\cD)$ and the number of flow modules $b(\\cD)$. We show that $\\cW(\\cD)\\geq \\frac{b(\\cD)+1}{2}$. 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