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We refine his methods in order to prove the optimal bound LL(ZB)\\le LL(FD) whenever D is abelian. We also improve Otokita's bound for non-abelian defect groups. As an application we classify the blocks B such that LL(ZB)\\ge |D|/2."},"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":"1607.06241","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-07-21T09:20:03Z","cross_cats_sorted":[],"title_canon_sha256":"1e654af76dd06a66da6c215ffe9486ee8552f985c4d7db2f6f2517025f28df96","abstract_canon_sha256":"a350a22937d348f61cb90673425d107f508f154c035300b9c1e1a81acd30c771"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:43.045108Z","signature_b64":"5QgdkTwifItEUQnr5c1DskC8mBBivthg3S7ssJaeuK1GeyxaQuIFU9/WoHLUodiCMPF3iYKPYcd6FshQtlA8CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ebb0061c78679291414d4658bfa4f83fb9ddf5acbdd274a24ea84481f4f2eeca","last_reissued_at":"2026-05-18T01:04:43.044630Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:43.044630Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Loewy lengths of centers of blocks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Benjamin Sambale, Burkhard K\\\"ulshammer","submitted_at":"2016-07-21T09:20:03Z","abstract_excerpt":"Let B be a block of a finite group with respect to an algebraically closed field F of characteristic p>0. 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