{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:HGLLYPBIOLOMDNOW3HFNDN3ROF","short_pith_number":"pith:HGLLYPBI","schema_version":"1.0","canonical_sha256":"3996bc3c2872dcc1b5d6d9cad1b7717143ed37230b31b3770b0d113426c22987","source":{"kind":"arxiv","id":"1506.05405","version":2},"attestation_state":"computed","paper":{"title":"Root subsystems of rank 2 hyperbolic root systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR","math.MP"],"primary_cat":"math-ph","authors_text":"Lisa Carbone, Matt Kownacki, Scott H. Murray, Sowmya Srinivasan","submitted_at":"2015-06-17T18:08:14Z","abstract_excerpt":"Let $\\Delta$ be a rank 2 hyperbolic root system. Then $\\Delta$ has generalized Cartan matrix $H(a,b)= \\left(\\begin{smallmatrix} ~2 & -b\\\\ -a & ~2 \\end{smallmatrix}\\right)$ indexed by $a,b\\in\\mathbb{Z}$ with $ab\\geq 5$. If $a\\neq b$, then $\\Delta$ is non-symmetric and is generated by one long simple root and one short simple root; whereas if $a= b$, $\\Delta$ is symmetric and is generated by two long simple roots. We prove that if $a\\neq b$, then $\\Delta$ contains an infinite family of symmetric rank 2 hyperbolic root subsystems $H(k,k)$ for certain $k\\geq 3$, generated by either two short or tw"},"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":"1506.05405","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2015-06-17T18:08:14Z","cross_cats_sorted":["math.CO","math.GR","math.MP"],"title_canon_sha256":"fda74448cd7d02f393545c9f8c6ee623f9bf0358e49ff5343f611c632f74ce4c","abstract_canon_sha256":"5eaaea6cf7a5add73789798f50540474485e4bc9e4e21143b0a799c545538c94"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:31.744198Z","signature_b64":"ZJbpDQ2d5XLJJxo2i02YnmXXTzCjaqZn93iBc5Mi98fkVkwWFdCkiH0cJneTnvhN8LNbq8XV4Yr+RRzEtvGsAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3996bc3c2872dcc1b5d6d9cad1b7717143ed37230b31b3770b0d113426c22987","last_reissued_at":"2026-05-18T00:56:31.743553Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:31.743553Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Root subsystems of rank 2 hyperbolic root systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GR","math.MP"],"primary_cat":"math-ph","authors_text":"Lisa Carbone, Matt Kownacki, Scott H. Murray, Sowmya Srinivasan","submitted_at":"2015-06-17T18:08:14Z","abstract_excerpt":"Let $\\Delta$ be a rank 2 hyperbolic root system. Then $\\Delta$ has generalized Cartan matrix $H(a,b)= \\left(\\begin{smallmatrix} ~2 & -b\\\\ -a & ~2 \\end{smallmatrix}\\right)$ indexed by $a,b\\in\\mathbb{Z}$ with $ab\\geq 5$. If $a\\neq b$, then $\\Delta$ is non-symmetric and is generated by one long simple root and one short simple root; whereas if $a= b$, $\\Delta$ is symmetric and is generated by two long simple roots. We prove that if $a\\neq b$, then $\\Delta$ contains an infinite family of symmetric rank 2 hyperbolic root subsystems $H(k,k)$ for certain $k\\geq 3$, generated by either two short or tw"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.05405","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1506.05405","created_at":"2026-05-18T00:56:31.743668+00:00"},{"alias_kind":"arxiv_version","alias_value":"1506.05405v2","created_at":"2026-05-18T00:56:31.743668+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.05405","created_at":"2026-05-18T00:56:31.743668+00:00"},{"alias_kind":"pith_short_12","alias_value":"HGLLYPBIOLOM","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_16","alias_value":"HGLLYPBIOLOMDNOW","created_at":"2026-05-18T12:29:25.134429+00:00"},{"alias_kind":"pith_short_8","alias_value":"HGLLYPBI","created_at":"2026-05-18T12:29:25.134429+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/HGLLYPBIOLOMDNOW3HFNDN3ROF","json":"https://pith.science/pith/HGLLYPBIOLOMDNOW3HFNDN3ROF.json","graph_json":"https://pith.science/api/pith-number/HGLLYPBIOLOMDNOW3HFNDN3ROF/graph.json","events_json":"https://pith.science/api/pith-number/HGLLYPBIOLOMDNOW3HFNDN3ROF/events.json","paper":"https://pith.science/paper/HGLLYPBI"},"agent_actions":{"view_html":"https://pith.science/pith/HGLLYPBIOLOMDNOW3HFNDN3ROF","download_json":"https://pith.science/pith/HGLLYPBIOLOMDNOW3HFNDN3ROF.json","view_paper":"https://pith.science/paper/HGLLYPBI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1506.05405&json=true","fetch_graph":"https://pith.science/api/pith-number/HGLLYPBIOLOMDNOW3HFNDN3ROF/graph.json","fetch_events":"https://pith.science/api/pith-number/HGLLYPBIOLOMDNOW3HFNDN3ROF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HGLLYPBIOLOMDNOW3HFNDN3ROF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HGLLYPBIOLOMDNOW3HFNDN3ROF/action/storage_attestation","attest_author":"https://pith.science/pith/HGLLYPBIOLOMDNOW3HFNDN3ROF/action/author_attestation","sign_citation":"https://pith.science/pith/HGLLYPBIOLOMDNOW3HFNDN3ROF/action/citation_signature","submit_replication":"https://pith.science/pith/HGLLYPBIOLOMDNOW3HFNDN3ROF/action/replication_record"}},"created_at":"2026-05-18T00:56:31.743668+00:00","updated_at":"2026-05-18T00:56:31.743668+00:00"}