{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:MD45GYVPBTWFJI6HQ3IXX7JNCC","short_pith_number":"pith:MD45GYVP","schema_version":"1.0","canonical_sha256":"60f9d362af0cec54a3c786d17bfd2d109c23ce55ca6e094c39f36b969173634d","source":{"kind":"arxiv","id":"1402.5871","version":1},"attestation_state":"computed","paper":{"title":"A characterisation of nilpotent blocks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RT","authors_text":"Gabriel Navarro, Markus Linckelmann, Radha Kessar","submitted_at":"2014-02-24T15:59:22Z","abstract_excerpt":"Let $B$ be a $p$-block of a finite group, and set $m=$ $\\sum \\chi(1)^2$, the sum taken over all height zero characters of $B$. Motivated by a result of M. Isaacs characterising $p$-nilpotent finite groups in terms of character degrees, we show that $B$ is nilpotent if and only if the exact power of $p$ dividing $m$ is equal to the $p$-part of $|G:P|^2|P:R|$, where $P$ is a defect group of $B$ and where $R$ is the focal subgroup of $P$ with respect to a fusion system $\\CF$ of $B$ on $P$. The proof involves the hyperfocal subalgebra $D$ of a source algebra of $B$. We conjecture that all ordinary"},"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":"1402.5871","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2014-02-24T15:59:22Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"ecf0895167da8854d98be71c4474ad0b11d348df0bb30c5bcd4aafb04e696750","abstract_canon_sha256":"8c84f8a4ddb35d344365b9ac66c60406d21252ced2060b6b4d6ecc609079ea22"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:13.396364Z","signature_b64":"9YLS7VAN1kWJx99mGSAeNbbtSbCs+8EhqPtuvYxNHaHvQQFmAYL7mh/qXq2/47EILeAXO8fSuFPCgTlR7xOrBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"60f9d362af0cec54a3c786d17bfd2d109c23ce55ca6e094c39f36b969173634d","last_reissued_at":"2026-05-18T02:58:13.395530Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:13.395530Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A characterisation of nilpotent blocks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.RT","authors_text":"Gabriel Navarro, Markus Linckelmann, Radha Kessar","submitted_at":"2014-02-24T15:59:22Z","abstract_excerpt":"Let $B$ be a $p$-block of a finite group, and set $m=$ $\\sum \\chi(1)^2$, the sum taken over all height zero characters of $B$. Motivated by a result of M. Isaacs characterising $p$-nilpotent finite groups in terms of character degrees, we show that $B$ is nilpotent if and only if the exact power of $p$ dividing $m$ is equal to the $p$-part of $|G:P|^2|P:R|$, where $P$ is a defect group of $B$ and where $R$ is the focal subgroup of $P$ with respect to a fusion system $\\CF$ of $B$ on $P$. The proof involves the hyperfocal subalgebra $D$ of a source algebra of $B$. We conjecture that all ordinary"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.5871","kind":"arxiv","version":1},"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":"1402.5871","created_at":"2026-05-18T02:58:13.395658+00:00"},{"alias_kind":"arxiv_version","alias_value":"1402.5871v1","created_at":"2026-05-18T02:58:13.395658+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.5871","created_at":"2026-05-18T02:58:13.395658+00:00"},{"alias_kind":"pith_short_12","alias_value":"MD45GYVPBTWF","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_16","alias_value":"MD45GYVPBTWFJI6H","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_8","alias_value":"MD45GYVP","created_at":"2026-05-18T12:28:38.356838+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/MD45GYVPBTWFJI6HQ3IXX7JNCC","json":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC.json","graph_json":"https://pith.science/api/pith-number/MD45GYVPBTWFJI6HQ3IXX7JNCC/graph.json","events_json":"https://pith.science/api/pith-number/MD45GYVPBTWFJI6HQ3IXX7JNCC/events.json","paper":"https://pith.science/paper/MD45GYVP"},"agent_actions":{"view_html":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC","download_json":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC.json","view_paper":"https://pith.science/paper/MD45GYVP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1402.5871&json=true","fetch_graph":"https://pith.science/api/pith-number/MD45GYVPBTWFJI6HQ3IXX7JNCC/graph.json","fetch_events":"https://pith.science/api/pith-number/MD45GYVPBTWFJI6HQ3IXX7JNCC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC/action/storage_attestation","attest_author":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC/action/author_attestation","sign_citation":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC/action/citation_signature","submit_replication":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC/action/replication_record"}},"created_at":"2026-05-18T02:58:13.395658+00:00","updated_at":"2026-05-18T02:58:13.395658+00:00"}