{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:E2WIZUESPAZXJ7UBA7ZB7LE2PI","short_pith_number":"pith:E2WIZUES","canonical_record":{"source":{"id":"1207.2244","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-07-10T07:15:18Z","cross_cats_sorted":[],"title_canon_sha256":"47c13b0e35c74233077deeddb732961fc35aaa297311fb19824da2127138c10c","abstract_canon_sha256":"eb2bd3e0f1dfdf675f7656792742e126e055e6d30d3da57d75bcddc208a82ff2"},"schema_version":"1.0"},"canonical_sha256":"26ac8cd092783374fe8107f21fac9a7a18cfc6548be0d7c721d5ca835652c2b0","source":{"kind":"arxiv","id":"1207.2244","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.2244","created_at":"2026-05-18T03:51:24Z"},{"alias_kind":"arxiv_version","alias_value":"1207.2244v1","created_at":"2026-05-18T03:51:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.2244","created_at":"2026-05-18T03:51:24Z"},{"alias_kind":"pith_short_12","alias_value":"E2WIZUESPAZX","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"E2WIZUESPAZXJ7UB","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"E2WIZUES","created_at":"2026-05-18T12:27:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:E2WIZUESPAZXJ7UBA7ZB7LE2PI","target":"record","payload":{"canonical_record":{"source":{"id":"1207.2244","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-07-10T07:15:18Z","cross_cats_sorted":[],"title_canon_sha256":"47c13b0e35c74233077deeddb732961fc35aaa297311fb19824da2127138c10c","abstract_canon_sha256":"eb2bd3e0f1dfdf675f7656792742e126e055e6d30d3da57d75bcddc208a82ff2"},"schema_version":"1.0"},"canonical_sha256":"26ac8cd092783374fe8107f21fac9a7a18cfc6548be0d7c721d5ca835652c2b0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:51:24.633505Z","signature_b64":"f2LT8ofxmpUMmJQNfYEIbn72SlMhXFeunWvR8gB96fMvSF04z3X9OpTWQ9EcUfy7EVbRv6/jJpAR1a4jjjDODA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"26ac8cd092783374fe8107f21fac9a7a18cfc6548be0d7c721d5ca835652c2b0","last_reissued_at":"2026-05-18T03:51:24.632951Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:51:24.632951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.2244","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:51:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vjRvl7GYa3UAsbzf5BoikdT9uQhP5r4HaiGoZexhuQi+x9m5+BgUvk/3JWtR9YbQCGqfoF8NO7sPL3OP/8sKAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:53:30.474570Z"},"content_sha256":"5e9683f6af6efed9cf49d0bd16e4bcd9ddb196a71b077400987fe60b4531cfc1","schema_version":"1.0","event_id":"sha256:5e9683f6af6efed9cf49d0bd16e4bcd9ddb196a71b077400987fe60b4531cfc1"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:E2WIZUESPAZXJ7UBA7ZB7LE2PI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weyl Groups Associated with Affine Reflection Systems of Type $A_1$ (Coxeter Type Defining Relations)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Mohammad Nikouei, Saeid Azam","submitted_at":"2012-07-10T07:15:18Z","abstract_excerpt":"In this paper, we offer a presentation for the Weyl group of an affine reflection system $R$ of type $A_1$ as well as a presentation for the so called hyperbolic Weyl group associated with an affine reflection system of type $A_1$. Applying these presentations to extended affine Weyl groups, and using a description of the center of the hyperbolic Weyl group, we also give a new finite presentation for an extended affine Weyl group of type $A_1$. Our presentation for the (hyperbolic) Weyl group of an affine reflection system of type $A_1$ is the first non-trivial presentation given in such a gen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2244","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":""},"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":"2026-05-18T03:51:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RCBi0NkBA96YLFMslk/Nw2NHQ3jFi60GlyPPSm2kfnZOWfUNQ9kgN+Y806sUsEF01x/zrgNwXGdmixxBeo6jDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:53:30.475275Z"},"content_sha256":"1cb2065a91a73922e5aa775795d01746a04cf1b54cbbdce79ce00202e32f2db1","schema_version":"1.0","event_id":"sha256:1cb2065a91a73922e5aa775795d01746a04cf1b54cbbdce79ce00202e32f2db1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E2WIZUESPAZXJ7UBA7ZB7LE2PI/bundle.json","state_url":"https://pith.science/pith/E2WIZUESPAZXJ7UBA7ZB7LE2PI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E2WIZUESPAZXJ7UBA7ZB7LE2PI/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":"2026-05-27T09:53:30Z","links":{"resolver":"https://pith.science/pith/E2WIZUESPAZXJ7UBA7ZB7LE2PI","bundle":"https://pith.science/pith/E2WIZUESPAZXJ7UBA7ZB7LE2PI/bundle.json","state":"https://pith.science/pith/E2WIZUESPAZXJ7UBA7ZB7LE2PI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E2WIZUESPAZXJ7UBA7ZB7LE2PI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:E2WIZUESPAZXJ7UBA7ZB7LE2PI","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":"eb2bd3e0f1dfdf675f7656792742e126e055e6d30d3da57d75bcddc208a82ff2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-07-10T07:15:18Z","title_canon_sha256":"47c13b0e35c74233077deeddb732961fc35aaa297311fb19824da2127138c10c"},"schema_version":"1.0","source":{"id":"1207.2244","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.2244","created_at":"2026-05-18T03:51:24Z"},{"alias_kind":"arxiv_version","alias_value":"1207.2244v1","created_at":"2026-05-18T03:51:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.2244","created_at":"2026-05-18T03:51:24Z"},{"alias_kind":"pith_short_12","alias_value":"E2WIZUESPAZX","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_16","alias_value":"E2WIZUESPAZXJ7UB","created_at":"2026-05-18T12:27:04Z"},{"alias_kind":"pith_short_8","alias_value":"E2WIZUES","created_at":"2026-05-18T12:27:04Z"}],"graph_snapshots":[{"event_id":"sha256:1cb2065a91a73922e5aa775795d01746a04cf1b54cbbdce79ce00202e32f2db1","target":"graph","created_at":"2026-05-18T03:51:24Z","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":"In this paper, we offer a presentation for the Weyl group of an affine reflection system $R$ of type $A_1$ as well as a presentation for the so called hyperbolic Weyl group associated with an affine reflection system of type $A_1$. Applying these presentations to extended affine Weyl groups, and using a description of the center of the hyperbolic Weyl group, we also give a new finite presentation for an extended affine Weyl group of type $A_1$. Our presentation for the (hyperbolic) Weyl group of an affine reflection system of type $A_1$ is the first non-trivial presentation given in such a gen","authors_text":"Mohammad Nikouei, Saeid Azam","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-07-10T07:15:18Z","title":"Weyl Groups Associated with Affine Reflection Systems of Type $A_1$ (Coxeter Type Defining Relations)"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2244","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":"sha256:5e9683f6af6efed9cf49d0bd16e4bcd9ddb196a71b077400987fe60b4531cfc1","target":"record","created_at":"2026-05-18T03:51:24Z","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":"eb2bd3e0f1dfdf675f7656792742e126e055e6d30d3da57d75bcddc208a82ff2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-07-10T07:15:18Z","title_canon_sha256":"47c13b0e35c74233077deeddb732961fc35aaa297311fb19824da2127138c10c"},"schema_version":"1.0","source":{"id":"1207.2244","kind":"arxiv","version":1}},"canonical_sha256":"26ac8cd092783374fe8107f21fac9a7a18cfc6548be0d7c721d5ca835652c2b0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"26ac8cd092783374fe8107f21fac9a7a18cfc6548be0d7c721d5ca835652c2b0","first_computed_at":"2026-05-18T03:51:24.632951Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:51:24.632951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"f2LT8ofxmpUMmJQNfYEIbn72SlMhXFeunWvR8gB96fMvSF04z3X9OpTWQ9EcUfy7EVbRv6/jJpAR1a4jjjDODA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:51:24.633505Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.2244","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5e9683f6af6efed9cf49d0bd16e4bcd9ddb196a71b077400987fe60b4531cfc1","sha256:1cb2065a91a73922e5aa775795d01746a04cf1b54cbbdce79ce00202e32f2db1"],"state_sha256":"57c4d3701e54b5ed8d60cf25d3a54241a9460713d40baff8a2f43290f79b9e2b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PCf7Gp3cFkBpL8sthihZKS4Y1mMNk882mCgxPRiWFG3b+5PTtOZNjLNIm5NsykYtkwHeat8N5U/dJbFbl5EIDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T09:53:30.478883Z","bundle_sha256":"aa129ba06c0f58b76e08c617428166dd9ca3f668e73424b61761b39692b14ede"}}