{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:F7CIIVSR4L2MLHUWEQOO7PQPKP","short_pith_number":"pith:F7CIIVSR","canonical_record":{"source":{"id":"1612.00736","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-02T16:29:15Z","cross_cats_sorted":[],"title_canon_sha256":"d2644a8a0b76a9b726aeefc53f7d13d7ec0c290874458910334b568705343f5d","abstract_canon_sha256":"3245fbbc7d422e7450f0647cbf66d007f1b410399afe207c20bbcc93905a9169"},"schema_version":"1.0"},"canonical_sha256":"2fc4845651e2f4c59e96241cefbe0f53e84bd61128d4ffbb2e06730ee4b173d5","source":{"kind":"arxiv","id":"1612.00736","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.00736","created_at":"2026-05-18T00:28:02Z"},{"alias_kind":"arxiv_version","alias_value":"1612.00736v2","created_at":"2026-05-18T00:28:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.00736","created_at":"2026-05-18T00:28:02Z"},{"alias_kind":"pith_short_12","alias_value":"F7CIIVSR4L2M","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"F7CIIVSR4L2MLHUW","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"F7CIIVSR","created_at":"2026-05-18T12:30:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:F7CIIVSR4L2MLHUWEQOO7PQPKP","target":"record","payload":{"canonical_record":{"source":{"id":"1612.00736","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-02T16:29:15Z","cross_cats_sorted":[],"title_canon_sha256":"d2644a8a0b76a9b726aeefc53f7d13d7ec0c290874458910334b568705343f5d","abstract_canon_sha256":"3245fbbc7d422e7450f0647cbf66d007f1b410399afe207c20bbcc93905a9169"},"schema_version":"1.0"},"canonical_sha256":"2fc4845651e2f4c59e96241cefbe0f53e84bd61128d4ffbb2e06730ee4b173d5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:02.585009Z","signature_b64":"JhTZsHmqihfw7TVFofu/YDxKU637jnrqZLYMc8HsXvR0HqiF7Wg3uqmqssA65zj6DV3h8Oe/WEFhqOdPmAKaDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2fc4845651e2f4c59e96241cefbe0f53e84bd61128d4ffbb2e06730ee4b173d5","last_reissued_at":"2026-05-18T00:28:02.584329Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:02.584329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.00736","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":"2026-05-18T00:28:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dVjfoM7auE7zvZ2mcb09ckd7OtuNgmtshSW8t04ejSZIDT75h9xuVJ5hdaBvVl5AGQMssJp/JAKfPJllOI16AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T08:14:13.916020Z"},"content_sha256":"5b95ecd10296db7e7e340d11cd8899823714d3ecdb6a1f92510d339239f629ba","schema_version":"1.0","event_id":"sha256:5b95ecd10296db7e7e340d11cd8899823714d3ecdb6a1f92510d339239f629ba"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:F7CIIVSR4L2MLHUWEQOO7PQPKP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Parabolic double cosets in Coxeter groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bridget E. Tenner, Matja\\v{z} Konvalinka, Sara C. Billey, T. Kyle Petersen, William Slofstra","submitted_at":"2016-12-02T16:29:15Z","abstract_excerpt":"Parabolic subgroups $W_I$ of Coxeter systems $(W,S)$, as well as their ordinary and double quotients $W / W_I$ and $W_I \\backslash W / W_J$, appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets $w W_I$, for $I \\subseteq S$, forms the Coxeter complex of $W$, and is well-studied. In this article we look at a less studied object: the set of all double cosets $W_I w W_J$ for $I, J \\subseteq S$. Double coset are not uniquely presented by triples $(I,w,J)$. W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.00736","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":"2026-05-18T00:28:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EjYYQcMMmrxwifHLH0Mj/CvmGBjMIcNQapkmxp1dnOLeQdY9wJkdMo0XTSXqHwP5vmsJirz2rIdrL49xEe8uAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T08:14:13.916579Z"},"content_sha256":"28893f288e392f44c9cb008cbff198caa7b4ca4fc16d9468720bfc21d4f8ac28","schema_version":"1.0","event_id":"sha256:28893f288e392f44c9cb008cbff198caa7b4ca4fc16d9468720bfc21d4f8ac28"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F7CIIVSR4L2MLHUWEQOO7PQPKP/bundle.json","state_url":"https://pith.science/pith/F7CIIVSR4L2MLHUWEQOO7PQPKP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F7CIIVSR4L2MLHUWEQOO7PQPKP/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":"2026-06-03T08:14:13Z","links":{"resolver":"https://pith.science/pith/F7CIIVSR4L2MLHUWEQOO7PQPKP","bundle":"https://pith.science/pith/F7CIIVSR4L2MLHUWEQOO7PQPKP/bundle.json","state":"https://pith.science/pith/F7CIIVSR4L2MLHUWEQOO7PQPKP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F7CIIVSR4L2MLHUWEQOO7PQPKP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:F7CIIVSR4L2MLHUWEQOO7PQPKP","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":"3245fbbc7d422e7450f0647cbf66d007f1b410399afe207c20bbcc93905a9169","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-02T16:29:15Z","title_canon_sha256":"d2644a8a0b76a9b726aeefc53f7d13d7ec0c290874458910334b568705343f5d"},"schema_version":"1.0","source":{"id":"1612.00736","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.00736","created_at":"2026-05-18T00:28:02Z"},{"alias_kind":"arxiv_version","alias_value":"1612.00736v2","created_at":"2026-05-18T00:28:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.00736","created_at":"2026-05-18T00:28:02Z"},{"alias_kind":"pith_short_12","alias_value":"F7CIIVSR4L2M","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_16","alias_value":"F7CIIVSR4L2MLHUW","created_at":"2026-05-18T12:30:15Z"},{"alias_kind":"pith_short_8","alias_value":"F7CIIVSR","created_at":"2026-05-18T12:30:15Z"}],"graph_snapshots":[{"event_id":"sha256:28893f288e392f44c9cb008cbff198caa7b4ca4fc16d9468720bfc21d4f8ac28","target":"graph","created_at":"2026-05-18T00:28:02Z","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":"Parabolic subgroups $W_I$ of Coxeter systems $(W,S)$, as well as their ordinary and double quotients $W / W_I$ and $W_I \\backslash W / W_J$, appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets $w W_I$, for $I \\subseteq S$, forms the Coxeter complex of $W$, and is well-studied. In this article we look at a less studied object: the set of all double cosets $W_I w W_J$ for $I, J \\subseteq S$. Double coset are not uniquely presented by triples $(I,w,J)$. W","authors_text":"Bridget E. Tenner, Matja\\v{z} Konvalinka, Sara C. Billey, T. Kyle Petersen, William Slofstra","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-02T16:29:15Z","title":"Parabolic double cosets in Coxeter groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.00736","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":"sha256:5b95ecd10296db7e7e340d11cd8899823714d3ecdb6a1f92510d339239f629ba","target":"record","created_at":"2026-05-18T00:28:02Z","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":"3245fbbc7d422e7450f0647cbf66d007f1b410399afe207c20bbcc93905a9169","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-02T16:29:15Z","title_canon_sha256":"d2644a8a0b76a9b726aeefc53f7d13d7ec0c290874458910334b568705343f5d"},"schema_version":"1.0","source":{"id":"1612.00736","kind":"arxiv","version":2}},"canonical_sha256":"2fc4845651e2f4c59e96241cefbe0f53e84bd61128d4ffbb2e06730ee4b173d5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2fc4845651e2f4c59e96241cefbe0f53e84bd61128d4ffbb2e06730ee4b173d5","first_computed_at":"2026-05-18T00:28:02.584329Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:28:02.584329Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JhTZsHmqihfw7TVFofu/YDxKU637jnrqZLYMc8HsXvR0HqiF7Wg3uqmqssA65zj6DV3h8Oe/WEFhqOdPmAKaDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:28:02.585009Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.00736","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5b95ecd10296db7e7e340d11cd8899823714d3ecdb6a1f92510d339239f629ba","sha256:28893f288e392f44c9cb008cbff198caa7b4ca4fc16d9468720bfc21d4f8ac28"],"state_sha256":"62e9caae853ed6ab6ab75f74a47ba3404e62bc632869c0cf9954dd4393a318b8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1raE7Y1AmioEZUi9UuUJXdkFFkoEOk8Tc28vGK9EZ1Oz/6CRMujZglbAu0Lf9js53jMW1cbaw2ESu+v2iNslAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T08:14:13.919211Z","bundle_sha256":"39ed9a78362e5e61d39898155a8f2eed36361f191366d0bbf25bc591da01a827"}}