{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:CSCY4GDM5XA3TDRHCF67XPTUYY","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":"8c2cdcc4934b2af8fb6c09a8504edcd7296d6751a6dc13c6dabec85dbcd645b5","cross_cats_sorted":["math.CO","math.GR","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-06-01T15:57:38Z","title_canon_sha256":"8f00a044a14cf1a4c320fe2645081c58f4ba21c4f554e1d1fd9343984a0963f7"},"schema_version":"1.0","source":{"id":"2606.02415","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.02415","created_at":"2026-06-02T03:04:58Z"},{"alias_kind":"arxiv_version","alias_value":"2606.02415v1","created_at":"2026-06-02T03:04:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.02415","created_at":"2026-06-02T03:04:58Z"},{"alias_kind":"pith_short_12","alias_value":"CSCY4GDM5XA3","created_at":"2026-06-02T03:04:58Z"},{"alias_kind":"pith_short_16","alias_value":"CSCY4GDM5XA3TDRH","created_at":"2026-06-02T03:04:58Z"},{"alias_kind":"pith_short_8","alias_value":"CSCY4GDM","created_at":"2026-06-02T03:04:58Z"}],"graph_snapshots":[{"event_id":"sha256:c7947ac025f62bc9b0207f0b61a70150c9cc9c33eef8d72720da719dde0d2714","target":"graph","created_at":"2026-06-02T03:04:58Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.02415/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this article, we study rational representations of $G=\\operatorname{GL}_2(q)$, where $q$ is a prime power. Let $\\rho$ be an irreducible representation of $G$ over $\\mathbb{Q}$. Then $\\rho$ affords the character \\[ \\Omega(\\chi)=m_{\\mathbb{Q}}(\\chi)\\sum_{\\sigma\\in\\operatorname{Gal}(\\mathbb{Q}(\\chi)/\\mathbb{Q})}\\chi^{\\sigma}, \\] for some irreducible complex character $\\chi$ of $G$, where $m_{\\mathbb{Q}}(\\chi)$ denotes the Schur index of $\\chi$ over $\\mathbb{Q}$, with the converse also holding. We obtain a combinatorial description for the counting of inequivalent irreducible $\\mathbb{Q}$-repre","authors_text":"Ram Karan Choudhary, Sunil Kumar Prajapati","cross_cats":["math.CO","math.GR","math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-06-01T15:57:38Z","title":"On rational representations and rational group algebra of $\\operatorname{GL}_2(q)$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02415","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:fa8933fe9394c1ed89d823b1bd7246aab2c1fe9fdbd15e0bd44ba55585f5ab12","target":"record","created_at":"2026-06-02T03:04:58Z","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":"8c2cdcc4934b2af8fb6c09a8504edcd7296d6751a6dc13c6dabec85dbcd645b5","cross_cats_sorted":["math.CO","math.GR","math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2026-06-01T15:57:38Z","title_canon_sha256":"8f00a044a14cf1a4c320fe2645081c58f4ba21c4f554e1d1fd9343984a0963f7"},"schema_version":"1.0","source":{"id":"2606.02415","kind":"arxiv","version":1}},"canonical_sha256":"14858e186cedc1b98e27117dfbbe74c6166738613e7d43d3ab7fe83086587eb4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14858e186cedc1b98e27117dfbbe74c6166738613e7d43d3ab7fe83086587eb4","first_computed_at":"2026-06-02T03:04:58.577365Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-02T03:04:58.577365Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OPX9G0OZXLUqe8+DiTUusetl/noGprom25TiFU3YNthlYrigvuwRotntouW4eW4HjgSAa9tAIWKoDvjiED9yCA==","signature_status":"signed_v1","signed_at":"2026-06-02T03:04:58.577705Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.02415","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fa8933fe9394c1ed89d823b1bd7246aab2c1fe9fdbd15e0bd44ba55585f5ab12","sha256:c7947ac025f62bc9b0207f0b61a70150c9cc9c33eef8d72720da719dde0d2714"],"state_sha256":"b73f7adcf10320024f249ec4ec85151f325ca66d4a2cd6919833319030772cb1"}