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We investigate the relationship between $T(G)$ and the number of twisted involutions $m_\\sigma = |\\{g \\in G \\mid \\sigma(g) = g^{-1}\\}|$ for an automorphism $\\sigma$. While it is known that $T(G) = m_e$ for the identity automorphism $e$ in certain cases (e.g., real characters), we analyze this relation for non-identity automorphisms of groups of order $p, 2p, p^2$. We prove that for the family of Dihedral groups $D_n$, the inequality $T(D_n) \\geq m_\\sigma$ holds for all $\\sigma \\in \\mathrm"},"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":"2605.22127","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GR","submitted_at":"2026-05-21T08:03:13Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"88b63b12a6c374691b4b0555bd8d8287bafef9348b19bc799c1b180873a9a722","abstract_canon_sha256":"63376d088f3026a1eda1ef287b5569753ab0fc5768eb5faafe93476a4ed11189"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-22T01:04:27.183453Z","signature_b64":"90sV7bHVvylB0TSgU4Jo4h4LFCfKtfehDWfv9gcifRmU8jnAjQwoj8ejFs6sFO6vVWtlBHcJPzMlR+gk//RaAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb76227e3591889651604424eeaf1837c11691fb011859d89c75f430e6ad3e7e","last_reissued_at":"2026-05-22T01:04:27.182784Z","signature_status":"signed_v1","first_computed_at":"2026-05-22T01:04:27.182784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Twisted Frobenius-Schur Indicators and Character Degree Sums in Dihedral Groups","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.GR","authors_text":"Ajay Kumar Shukla, Rahul Dixit, Venkata Subbaiah Yerrapati","submitted_at":"2026-05-21T08:03:13Z","abstract_excerpt":"Let $G$ be a finite group and $T(G)$ be the sum of the degrees of its irreducible complex representations. 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We prove that for the family of Dihedral groups $D_n$, the inequality $T(D_n) \\geq m_\\sigma$ holds for all $\\sigma \\in \\mathrm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.22127","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.22127/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2605.22127","created_at":"2026-05-22T01:04:27.182891+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.22127v1","created_at":"2026-05-22T01:04:27.182891+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.22127","created_at":"2026-05-22T01:04:27.182891+00:00"},{"alias_kind":"pith_short_12","alias_value":"5N3CE7RVSGEJ","created_at":"2026-05-22T01:04:27.182891+00:00"},{"alias_kind":"pith_short_16","alias_value":"5N3CE7RVSGEJMULA","created_at":"2026-05-22T01:04:27.182891+00:00"},{"alias_kind":"pith_short_8","alias_value":"5N3CE7RV","created_at":"2026-05-22T01:04:27.182891+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/5N3CE7RVSGEJMULAIQSO5LYYG7","json":"https://pith.science/pith/5N3CE7RVSGEJMULAIQSO5LYYG7.json","graph_json":"https://pith.science/api/pith-number/5N3CE7RVSGEJMULAIQSO5LYYG7/graph.json","events_json":"https://pith.science/api/pith-number/5N3CE7RVSGEJMULAIQSO5LYYG7/events.json","paper":"https://pith.science/paper/5N3CE7RV"},"agent_actions":{"view_html":"https://pith.science/pith/5N3CE7RVSGEJMULAIQSO5LYYG7","download_json":"https://pith.science/pith/5N3CE7RVSGEJMULAIQSO5LYYG7.json","view_paper":"https://pith.science/paper/5N3CE7RV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.22127&json=true","fetch_graph":"https://pith.science/api/pith-number/5N3CE7RVSGEJMULAIQSO5LYYG7/graph.json","fetch_events":"https://pith.science/api/pith-number/5N3CE7RVSGEJMULAIQSO5LYYG7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5N3CE7RVSGEJMULAIQSO5LYYG7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5N3CE7RVSGEJMULAIQSO5LYYG7/action/storage_attestation","attest_author":"https://pith.science/pith/5N3CE7RVSGEJMULAIQSO5LYYG7/action/author_attestation","sign_citation":"https://pith.science/pith/5N3CE7RVSGEJMULAIQSO5LYYG7/action/citation_signature","submit_replication":"https://pith.science/pith/5N3CE7RVSGEJMULAIQSO5LYYG7/action/replication_record"}},"created_at":"2026-05-22T01:04:27.182891+00:00","updated_at":"2026-05-22T01:04:27.182891+00:00"}