{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:WLARNPYY2V2ZVPADIPVYFSF237","short_pith_number":"pith:WLARNPYY","canonical_record":{"source":{"id":"1503.05326","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-03-18T10:11:35Z","cross_cats_sorted":[],"title_canon_sha256":"a5932e87d466950b5128eea0a1d0654c0a313ead946a63a533fa89966e9d3ee9","abstract_canon_sha256":"f99e4adfed110abaf902897bf73f3b92a165230402e08471101512f5ba401a6f"},"schema_version":"1.0"},"canonical_sha256":"b2c116bf18d5759abc0343eb82c8badff0b3d61c59af0003700d111845ab410c","source":{"kind":"arxiv","id":"1503.05326","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.05326","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"arxiv_version","alias_value":"1503.05326v2","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.05326","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"pith_short_12","alias_value":"WLARNPYY2V2Z","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WLARNPYY2V2ZVPAD","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WLARNPYY","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:WLARNPYY2V2ZVPADIPVYFSF237","target":"record","payload":{"canonical_record":{"source":{"id":"1503.05326","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-03-18T10:11:35Z","cross_cats_sorted":[],"title_canon_sha256":"a5932e87d466950b5128eea0a1d0654c0a313ead946a63a533fa89966e9d3ee9","abstract_canon_sha256":"f99e4adfed110abaf902897bf73f3b92a165230402e08471101512f5ba401a6f"},"schema_version":"1.0"},"canonical_sha256":"b2c116bf18d5759abc0343eb82c8badff0b3d61c59af0003700d111845ab410c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:34:41.960755Z","signature_b64":"Dtiy1OMOUYEix3wNZel3z6P88l7FFTzkA1CS+aQDa4nN7MA/gcPdhLtVAcDWXewgOgP8meVWgFcGgJpmXY49BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2c116bf18d5759abc0343eb82c8badff0b3d61c59af0003700d111845ab410c","last_reissued_at":"2026-05-18T01:34:41.960108Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:34:41.960108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.05326","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":"2026-05-18T01:34:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AEHihDfxZzyX/wFuphk4fLdbKpn4sN8iWdt2QCcLzKriXoNMD3q9lTk6p0fLpQSRsWzzzSV2Cx2vNGS57cp4CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T09:37:26.478260Z"},"content_sha256":"01c014ca9d4464256af8d485b56b7684c6819b738347e8a467d9cdd485157c42","schema_version":"1.0","event_id":"sha256:01c014ca9d4464256af8d485b56b7684c6819b738347e8a467d9cdd485157c42"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:WLARNPYY2V2ZVPADIPVYFSF237","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Maximal length elements of excess zero in finite Coxeter groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Peter J. Rowley, Sarah B. Hart","submitted_at":"2015-03-18T10:11:35Z","abstract_excerpt":"The excess of an element $w$ of a finite Coxeter group $W$ is the minimal value of $l(x) + l(y) - l(w)$, where $x$, $y$ are elements of $W$ such that $x^2 = y^2 = 1$ and $w = xy$. Every element of a finite Coxeter group is either an involution or the product of two involutions, so the concept of excess is well defined. It can be extended to strongly real classes of infinite Coxeter groups. Earlier work by the authors showed that every conjugacy class of a finite Coxeter group contains an element of minimal length and excess zero. The current paper shows that each conjugacy class also contains "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.05326","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":"2026-05-18T01:34:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JqgmBS3wRwso6L+5t19SPU7ZzEP+zDK0u3SCdyJefujLsZTYzhWvl6czbKIzb63fcUFRq+DZM8CgSEGEuJZ+Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T09:37:26.478923Z"},"content_sha256":"3121494445088321a98bbfa4ae0b85b77c89b0127fe78e375f43f0163a69615a","schema_version":"1.0","event_id":"sha256:3121494445088321a98bbfa4ae0b85b77c89b0127fe78e375f43f0163a69615a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WLARNPYY2V2ZVPADIPVYFSF237/bundle.json","state_url":"https://pith.science/pith/WLARNPYY2V2ZVPADIPVYFSF237/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WLARNPYY2V2ZVPADIPVYFSF237/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":"2026-06-22T09:37:26Z","links":{"resolver":"https://pith.science/pith/WLARNPYY2V2ZVPADIPVYFSF237","bundle":"https://pith.science/pith/WLARNPYY2V2ZVPADIPVYFSF237/bundle.json","state":"https://pith.science/pith/WLARNPYY2V2ZVPADIPVYFSF237/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WLARNPYY2V2ZVPADIPVYFSF237/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:WLARNPYY2V2ZVPADIPVYFSF237","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":"f99e4adfed110abaf902897bf73f3b92a165230402e08471101512f5ba401a6f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-03-18T10:11:35Z","title_canon_sha256":"a5932e87d466950b5128eea0a1d0654c0a313ead946a63a533fa89966e9d3ee9"},"schema_version":"1.0","source":{"id":"1503.05326","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.05326","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"arxiv_version","alias_value":"1503.05326v2","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.05326","created_at":"2026-05-18T01:34:41Z"},{"alias_kind":"pith_short_12","alias_value":"WLARNPYY2V2Z","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WLARNPYY2V2ZVPAD","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WLARNPYY","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:3121494445088321a98bbfa4ae0b85b77c89b0127fe78e375f43f0163a69615a","target":"graph","created_at":"2026-05-18T01:34:41Z","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":"The excess of an element $w$ of a finite Coxeter group $W$ is the minimal value of $l(x) + l(y) - l(w)$, where $x$, $y$ are elements of $W$ such that $x^2 = y^2 = 1$ and $w = xy$. Every element of a finite Coxeter group is either an involution or the product of two involutions, so the concept of excess is well defined. It can be extended to strongly real classes of infinite Coxeter groups. Earlier work by the authors showed that every conjugacy class of a finite Coxeter group contains an element of minimal length and excess zero. The current paper shows that each conjugacy class also contains ","authors_text":"Peter J. Rowley, Sarah B. Hart","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-03-18T10:11:35Z","title":"Maximal length elements of excess zero in finite Coxeter groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.05326","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":"sha256:01c014ca9d4464256af8d485b56b7684c6819b738347e8a467d9cdd485157c42","target":"record","created_at":"2026-05-18T01:34:41Z","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":"f99e4adfed110abaf902897bf73f3b92a165230402e08471101512f5ba401a6f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-03-18T10:11:35Z","title_canon_sha256":"a5932e87d466950b5128eea0a1d0654c0a313ead946a63a533fa89966e9d3ee9"},"schema_version":"1.0","source":{"id":"1503.05326","kind":"arxiv","version":2}},"canonical_sha256":"b2c116bf18d5759abc0343eb82c8badff0b3d61c59af0003700d111845ab410c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b2c116bf18d5759abc0343eb82c8badff0b3d61c59af0003700d111845ab410c","first_computed_at":"2026-05-18T01:34:41.960108Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:34:41.960108Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Dtiy1OMOUYEix3wNZel3z6P88l7FFTzkA1CS+aQDa4nN7MA/gcPdhLtVAcDWXewgOgP8meVWgFcGgJpmXY49BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:34:41.960755Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.05326","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:01c014ca9d4464256af8d485b56b7684c6819b738347e8a467d9cdd485157c42","sha256:3121494445088321a98bbfa4ae0b85b77c89b0127fe78e375f43f0163a69615a"],"state_sha256":"2d95d71faa6d2ea53a0bf0b7ceddc3466a03ff9a77692502fdc13609d5b92128"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H8X2wLE5001YFXfL0QlSTurxLidFriok2E/TD01QumAosjSm6oWYoBhOY9isJXGRbfk6WR4wIorTBuVQzRZsAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T09:37:26.481930Z","bundle_sha256":"e7a5f1a36edf306b37f4dd6d4fee7c646e99ba099e34077eec750e48209d23a1"}}