{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:MD45GYVPBTWFJI6HQ3IXX7JNCC","short_pith_number":"pith:MD45GYVP","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"},"canonical_sha256":"60f9d362af0cec54a3c786d17bfd2d109c23ce55ca6e094c39f36b969173634d","source":{"kind":"arxiv","id":"1402.5871","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.5871","created_at":"2026-05-18T02:58:13Z"},{"alias_kind":"arxiv_version","alias_value":"1402.5871v1","created_at":"2026-05-18T02:58:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.5871","created_at":"2026-05-18T02:58:13Z"},{"alias_kind":"pith_short_12","alias_value":"MD45GYVPBTWF","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MD45GYVPBTWFJI6H","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MD45GYVP","created_at":"2026-05-18T12:28:38Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:MD45GYVPBTWFJI6HQ3IXX7JNCC","target":"record","payload":{"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"},"canonical_sha256":"60f9d362af0cec54a3c786d17bfd2d109c23ce55ca6e094c39f36b969173634d","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"},"source_kind":"arxiv","source_id":"1402.5871","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y33nY6mmQHCKuafIiSxZYoPVPStc076H5fg8KYFXs7OfKSmZeNKg22PYMwO9mN4u8wGsPZ+46TyicTQziNAxDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T16:46:09.340356Z"},"content_sha256":"1ddcfced34690490237a376e7ee726f0acc6b73810d847871073ace17a4f8c52","schema_version":"1.0","event_id":"sha256:1ddcfced34690490237a376e7ee726f0acc6b73810d847871073ace17a4f8c52"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:MD45GYVPBTWFJI6HQ3IXX7JNCC","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:58:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FgTmtAcK95RU8udmtYLJKdkjG5qOENpnHtuLHyrqfBrXW1GttJP1k0fbHwUPSAQcV6SKYe8QxQynGuOVS8GYDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T16:46:09.340709Z"},"content_sha256":"46fbb5102d0b6eb33a4895026ea22b6c479924dafa43a89510280abf80db34b1","schema_version":"1.0","event_id":"sha256:46fbb5102d0b6eb33a4895026ea22b6c479924dafa43a89510280abf80db34b1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC/bundle.json","state_url":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-03T16:46:09Z","links":{"resolver":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC","bundle":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC/bundle.json","state":"https://pith.science/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MD45GYVPBTWFJI6HQ3IXX7JNCC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:MD45GYVPBTWFJI6HQ3IXX7JNCC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8c84f8a4ddb35d344365b9ac66c60406d21252ced2060b6b4d6ecc609079ea22","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2014-02-24T15:59:22Z","title_canon_sha256":"ecf0895167da8854d98be71c4474ad0b11d348df0bb30c5bcd4aafb04e696750"},"schema_version":"1.0","source":{"id":"1402.5871","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.5871","created_at":"2026-05-18T02:58:13Z"},{"alias_kind":"arxiv_version","alias_value":"1402.5871v1","created_at":"2026-05-18T02:58:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.5871","created_at":"2026-05-18T02:58:13Z"},{"alias_kind":"pith_short_12","alias_value":"MD45GYVPBTWF","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"MD45GYVPBTWFJI6H","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"MD45GYVP","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:46fbb5102d0b6eb33a4895026ea22b6c479924dafa43a89510280abf80db34b1","target":"graph","created_at":"2026-05-18T02:58:13Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Gabriel Navarro, Markus Linckelmann, Radha Kessar","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2014-02-24T15:59:22Z","title":"A characterisation of nilpotent blocks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.5871","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1ddcfced34690490237a376e7ee726f0acc6b73810d847871073ace17a4f8c52","target":"record","created_at":"2026-05-18T02:58:13Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8c84f8a4ddb35d344365b9ac66c60406d21252ced2060b6b4d6ecc609079ea22","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2014-02-24T15:59:22Z","title_canon_sha256":"ecf0895167da8854d98be71c4474ad0b11d348df0bb30c5bcd4aafb04e696750"},"schema_version":"1.0","source":{"id":"1402.5871","kind":"arxiv","version":1}},"canonical_sha256":"60f9d362af0cec54a3c786d17bfd2d109c23ce55ca6e094c39f36b969173634d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"60f9d362af0cec54a3c786d17bfd2d109c23ce55ca6e094c39f36b969173634d","first_computed_at":"2026-05-18T02:58:13.395530Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:13.395530Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9YLS7VAN1kWJx99mGSAeNbbtSbCs+8EhqPtuvYxNHaHvQQFmAYL7mh/qXq2/47EILeAXO8fSuFPCgTlR7xOrBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:13.396364Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.5871","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1ddcfced34690490237a376e7ee726f0acc6b73810d847871073ace17a4f8c52","sha256:46fbb5102d0b6eb33a4895026ea22b6c479924dafa43a89510280abf80db34b1"],"state_sha256":"f3def35660cb8a1b1301c8eca34ba0fa0cba2932a39ea2ac93bbed15e16771a2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AHuPFgToopCZ4MU92hrwJFbGNUJMNjuxuF2N1NEJ2sHQLRWROpht+PkchY2hzuil8DzLGtiPaLajvyFuraU0Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T16:46:09.343202Z","bundle_sha256":"3d72c0ad9a7d6a115abe723d410d4daa39cdf50e22b3b7498bab7bebfe7c44a3"}}