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We prove that a symmetric $FG$-module $V$ is hyperbolic if and only if the restriction $V_D$ of $V$ to every $F$-special subgroup $D$ of $G$ is hyperbolic, and also, if and on"},"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":"1409.3639","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-09-12T02:58:50Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"703b130035e3a7c8668164b9fc1e204833cae089b8396fd4939ac606162c29d4","abstract_canon_sha256":"33d04ffb495f182abc4c9adae7e46e2df083f79b17d1e7ccaf900093817b08f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:00.070451Z","signature_b64":"sey7IfoUvEJJAKSoh4g9MrBMTbdI0wAwV9ramPyRokOhs2/MQsHn6NahaBXqhV2GK7cgOEHwNr3mBkgjE3pYDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"467224c7862371167924d65847ef571c5fe1d33b648865fe73fe20399745dd72","last_reissued_at":"2026-05-18T02:43:00.069927Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:00.069927Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hyperbolic Modules of Finite Group Algebras over Finite Fields of Characteristic Two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.GR","authors_text":"Ping Jin, Yun Fan","submitted_at":"2014-09-12T02:58:50Z","abstract_excerpt":"Let $G$ be a finite group and let $F$ be a finite field of characteristic $2$. 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