{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:SSXOMFSGCDPKHTOQOWFN6APXGM","short_pith_number":"pith:SSXOMFSG","canonical_record":{"source":{"id":"1402.7046","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-02-27T20:01:16Z","cross_cats_sorted":[],"title_canon_sha256":"1449e4987da3f0f07c25373e84ea0adb021241d5144ad543f4194e554778f7b6","abstract_canon_sha256":"27448dae6052af2735288e43171bdf8dec77f468a346f758861170ab135b6f53"},"schema_version":"1.0"},"canonical_sha256":"94aee6164610dea3cdd0758adf01f7330d4c52e033a0022a931ff043abb0a8d0","source":{"kind":"arxiv","id":"1402.7046","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.7046","created_at":"2026-05-18T02:50:33Z"},{"alias_kind":"arxiv_version","alias_value":"1402.7046v2","created_at":"2026-05-18T02:50:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.7046","created_at":"2026-05-18T02:50:33Z"},{"alias_kind":"pith_short_12","alias_value":"SSXOMFSGCDPK","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"SSXOMFSGCDPKHTOQ","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"SSXOMFSG","created_at":"2026-05-18T12:28:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:SSXOMFSGCDPKHTOQOWFN6APXGM","target":"record","payload":{"canonical_record":{"source":{"id":"1402.7046","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-02-27T20:01:16Z","cross_cats_sorted":[],"title_canon_sha256":"1449e4987da3f0f07c25373e84ea0adb021241d5144ad543f4194e554778f7b6","abstract_canon_sha256":"27448dae6052af2735288e43171bdf8dec77f468a346f758861170ab135b6f53"},"schema_version":"1.0"},"canonical_sha256":"94aee6164610dea3cdd0758adf01f7330d4c52e033a0022a931ff043abb0a8d0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:33.667491Z","signature_b64":"+BCwAIZ/FBcNDc62a0r5+H/W6oZHifsRzd98fktHSPk6nRQB2PsKIkZOnim3Ico1nEnlCLlsX4oymaUM2yQVBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94aee6164610dea3cdd0758adf01f7330d4c52e033a0022a931ff043abb0a8d0","last_reissued_at":"2026-05-18T02:50:33.666864Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:33.666864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1402.7046","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:50:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IHbz6/ozqSsAff3sJQXUczvffZKnlkU0aX9tuvgQxSp9BbtZyi3Ta/pgVoDrjr7hDL4PFIrVuljnx5z1cgAjBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T02:00:15.470411Z"},"content_sha256":"36f8893b498b52d8c5809917438c20ebd8a50b02692f267f5bbad652388246d3","schema_version":"1.0","event_id":"sha256:36f8893b498b52d8c5809917438c20ebd8a50b02692f267f5bbad652388246d3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:SSXOMFSGCDPKHTOQOWFN6APXGM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Endotrivial Modules for the General Linear Group in a Nondefining Characteristic","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Daniel K. Nakano, Jon F. Carlson, Nadia Mazza","submitted_at":"2014-02-27T20:01:16Z","abstract_excerpt":"Suppose that $G$ is a finite group such that $\\operatorname{SL}(n,q)\\subseteq G \\subseteq \\operatorname{GL}(n,q)$, and that $Z$ is a central subgroup of $G$. Let $T(G/Z)$ be the abelian group of equivalence classes of endotrivial $k(G/Z)$-modules, where $k$ is an algebraically closed field of characteristic~$p$ not dividing $q$. We show that the torsion free rank of $T(G/Z)$ is at most one, and we determine $T(G/Z)$ in the case that the Sylow $p$-subgroup of $G$ is abelian and nontrivial. The proofs for the torsion subgroup of $T(G/Z)$ use the theory of Young modules for $\\operatorname{GL}(n,q"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.7046","kind":"arxiv","version":2},"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":"2026-05-18T02:50:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LJOfHoqAsJSp9Vrb4uNmiVuL1+qU72OS4Mnrz0SF2Tm9mx2nB5imsII0/++zvXdQGUCAdBM+sDSznhRjgrarBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T02:00:15.470768Z"},"content_sha256":"32ee58efcfb18720382b7e075bdd97a29fa41cb122e4631cf2a43571dac32e37","schema_version":"1.0","event_id":"sha256:32ee58efcfb18720382b7e075bdd97a29fa41cb122e4631cf2a43571dac32e37"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SSXOMFSGCDPKHTOQOWFN6APXGM/bundle.json","state_url":"https://pith.science/pith/SSXOMFSGCDPKHTOQOWFN6APXGM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SSXOMFSGCDPKHTOQOWFN6APXGM/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":"2026-06-21T02:00:15Z","links":{"resolver":"https://pith.science/pith/SSXOMFSGCDPKHTOQOWFN6APXGM","bundle":"https://pith.science/pith/SSXOMFSGCDPKHTOQOWFN6APXGM/bundle.json","state":"https://pith.science/pith/SSXOMFSGCDPKHTOQOWFN6APXGM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SSXOMFSGCDPKHTOQOWFN6APXGM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:SSXOMFSGCDPKHTOQOWFN6APXGM","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":"27448dae6052af2735288e43171bdf8dec77f468a346f758861170ab135b6f53","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-02-27T20:01:16Z","title_canon_sha256":"1449e4987da3f0f07c25373e84ea0adb021241d5144ad543f4194e554778f7b6"},"schema_version":"1.0","source":{"id":"1402.7046","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1402.7046","created_at":"2026-05-18T02:50:33Z"},{"alias_kind":"arxiv_version","alias_value":"1402.7046v2","created_at":"2026-05-18T02:50:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1402.7046","created_at":"2026-05-18T02:50:33Z"},{"alias_kind":"pith_short_12","alias_value":"SSXOMFSGCDPK","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"SSXOMFSGCDPKHTOQ","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"SSXOMFSG","created_at":"2026-05-18T12:28:49Z"}],"graph_snapshots":[{"event_id":"sha256:32ee58efcfb18720382b7e075bdd97a29fa41cb122e4631cf2a43571dac32e37","target":"graph","created_at":"2026-05-18T02:50:33Z","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":"Suppose that $G$ is a finite group such that $\\operatorname{SL}(n,q)\\subseteq G \\subseteq \\operatorname{GL}(n,q)$, and that $Z$ is a central subgroup of $G$. Let $T(G/Z)$ be the abelian group of equivalence classes of endotrivial $k(G/Z)$-modules, where $k$ is an algebraically closed field of characteristic~$p$ not dividing $q$. We show that the torsion free rank of $T(G/Z)$ is at most one, and we determine $T(G/Z)$ in the case that the Sylow $p$-subgroup of $G$ is abelian and nontrivial. The proofs for the torsion subgroup of $T(G/Z)$ use the theory of Young modules for $\\operatorname{GL}(n,q","authors_text":"Daniel K. Nakano, Jon F. Carlson, Nadia Mazza","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-02-27T20:01:16Z","title":"Endotrivial Modules for the General Linear Group in a Nondefining Characteristic"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.7046","kind":"arxiv","version":2},"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":"sha256:36f8893b498b52d8c5809917438c20ebd8a50b02692f267f5bbad652388246d3","target":"record","created_at":"2026-05-18T02:50:33Z","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":"27448dae6052af2735288e43171bdf8dec77f468a346f758861170ab135b6f53","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-02-27T20:01:16Z","title_canon_sha256":"1449e4987da3f0f07c25373e84ea0adb021241d5144ad543f4194e554778f7b6"},"schema_version":"1.0","source":{"id":"1402.7046","kind":"arxiv","version":2}},"canonical_sha256":"94aee6164610dea3cdd0758adf01f7330d4c52e033a0022a931ff043abb0a8d0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"94aee6164610dea3cdd0758adf01f7330d4c52e033a0022a931ff043abb0a8d0","first_computed_at":"2026-05-18T02:50:33.666864Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:50:33.666864Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+BCwAIZ/FBcNDc62a0r5+H/W6oZHifsRzd98fktHSPk6nRQB2PsKIkZOnim3Ico1nEnlCLlsX4oymaUM2yQVBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:50:33.667491Z","signed_message":"canonical_sha256_bytes"},"source_id":"1402.7046","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:36f8893b498b52d8c5809917438c20ebd8a50b02692f267f5bbad652388246d3","sha256:32ee58efcfb18720382b7e075bdd97a29fa41cb122e4631cf2a43571dac32e37"],"state_sha256":"bc9b271e0613403b4c95f2c1ddcfffef89ff96846d08914c59d3e52cec93b9f2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3dAAab9fGp8U8X21oXwaLpOliw8AmqZ4SowT352wWg+KAfGi6BTFs85GVwuvWaH6NieoKzGfphUQP87Zm6/cAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T02:00:15.472719Z","bundle_sha256":"d690fbb8e07142f19c8d49ef1e1ba1781b8af225842c663f39a3395971ca49e8"}}