{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:GKHHQMEIG64NP56L67QLRNI4JX","short_pith_number":"pith:GKHHQMEI","canonical_record":{"source":{"id":"1112.0824","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-12-05T03:21:51Z","cross_cats_sorted":[],"title_canon_sha256":"d21abfd4d6bdd533c09fa666159ff5d712a5d55b5c394897117db9381d445a70","abstract_canon_sha256":"21504372e2457e05d7943440cc1a95d5c4a301701423535b0584c76869053039"},"schema_version":"1.0"},"canonical_sha256":"328e78308837b8d7f7cbf7e0b8b51c4dd2eb31c0ce8462584266576cfad445c9","source":{"kind":"arxiv","id":"1112.0824","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.0824","created_at":"2026-05-17T23:53:19Z"},{"alias_kind":"arxiv_version","alias_value":"1112.0824v1","created_at":"2026-05-17T23:53:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.0824","created_at":"2026-05-17T23:53:19Z"},{"alias_kind":"pith_short_12","alias_value":"GKHHQMEIG64N","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"GKHHQMEIG64NP56L","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"GKHHQMEI","created_at":"2026-05-18T12:26:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:GKHHQMEIG64NP56L67QLRNI4JX","target":"record","payload":{"canonical_record":{"source":{"id":"1112.0824","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-12-05T03:21:51Z","cross_cats_sorted":[],"title_canon_sha256":"d21abfd4d6bdd533c09fa666159ff5d712a5d55b5c394897117db9381d445a70","abstract_canon_sha256":"21504372e2457e05d7943440cc1a95d5c4a301701423535b0584c76869053039"},"schema_version":"1.0"},"canonical_sha256":"328e78308837b8d7f7cbf7e0b8b51c4dd2eb31c0ce8462584266576cfad445c9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:19.087055Z","signature_b64":"xKzJFcCOqcLoIZo+c3wFOZKvC+4vXLWirkdmHh6I6vWOIzb1/miI1wVf1ogt/ErL2X4un9GgXKPjDt9zf6guDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"328e78308837b8d7f7cbf7e0b8b51c4dd2eb31c0ce8462584266576cfad445c9","last_reissued_at":"2026-05-17T23:53:19.086433Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:19.086433Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1112.0824","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":"2026-05-17T23:53:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8znpl1xWTzPYyPbrL+0t/B0d0ryYpBkFfO4c4+nGd4Zj+TQo6KEoOgWRF50WqxdXMJbY0hOWupTBJOCrFoW9Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T01:09:53.683817Z"},"content_sha256":"837952b2f3e7d97c62a9b73f0e40512d9ecf4185ebe4eec9c3b85a946e94d38e","schema_version":"1.0","event_id":"sha256:837952b2f3e7d97c62a9b73f0e40512d9ecf4185ebe4eec9c3b85a946e94d38e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:GKHHQMEIG64NP56L67QLRNI4JX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Minimal length elements of extended affine Coxeter groups, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Sian Nie, Xuhua He","submitted_at":"2011-12-05T03:21:51Z","abstract_excerpt":"Let $W$ be an extended affine Weyl group. We prove that minimal length elements $w_{\\co}$ of any conjugacy class $\\co$ of $W$ satisfy some special properties, generalizing results of Geck and Pfeiffer \\cite{GP} on finite Weyl groups. We then introduce the \"class polynomials\" for affine Hecke algebra $H$ and prove that $T_{w_\\co}$, where $\\co$ runs over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H, H]$. We also classify the conjugacy classes satisfying a generalization of Lusztig's conjecture \\cite{L4}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.0824","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":"2026-05-17T23:53:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zBH99Smi1iD+thySSwy+fnjLjie5WmzM8IpWYUc2pZhMNg0Ctm6p9YeIQbwXj9785SZxLMncvkvy7e10Y6hSCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T01:09:53.684155Z"},"content_sha256":"79084a7244d21f8fa3c564f348e108d84483bc89565a92e3d8c8c677829e3643","schema_version":"1.0","event_id":"sha256:79084a7244d21f8fa3c564f348e108d84483bc89565a92e3d8c8c677829e3643"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GKHHQMEIG64NP56L67QLRNI4JX/bundle.json","state_url":"https://pith.science/pith/GKHHQMEIG64NP56L67QLRNI4JX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GKHHQMEIG64NP56L67QLRNI4JX/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":"2026-06-09T01:09:53Z","links":{"resolver":"https://pith.science/pith/GKHHQMEIG64NP56L67QLRNI4JX","bundle":"https://pith.science/pith/GKHHQMEIG64NP56L67QLRNI4JX/bundle.json","state":"https://pith.science/pith/GKHHQMEIG64NP56L67QLRNI4JX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GKHHQMEIG64NP56L67QLRNI4JX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:GKHHQMEIG64NP56L67QLRNI4JX","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":"21504372e2457e05d7943440cc1a95d5c4a301701423535b0584c76869053039","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-12-05T03:21:51Z","title_canon_sha256":"d21abfd4d6bdd533c09fa666159ff5d712a5d55b5c394897117db9381d445a70"},"schema_version":"1.0","source":{"id":"1112.0824","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1112.0824","created_at":"2026-05-17T23:53:19Z"},{"alias_kind":"arxiv_version","alias_value":"1112.0824v1","created_at":"2026-05-17T23:53:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.0824","created_at":"2026-05-17T23:53:19Z"},{"alias_kind":"pith_short_12","alias_value":"GKHHQMEIG64N","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"GKHHQMEIG64NP56L","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"GKHHQMEI","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:79084a7244d21f8fa3c564f348e108d84483bc89565a92e3d8c8c677829e3643","target":"graph","created_at":"2026-05-17T23:53:19Z","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":"Let $W$ be an extended affine Weyl group. We prove that minimal length elements $w_{\\co}$ of any conjugacy class $\\co$ of $W$ satisfy some special properties, generalizing results of Geck and Pfeiffer \\cite{GP} on finite Weyl groups. We then introduce the \"class polynomials\" for affine Hecke algebra $H$ and prove that $T_{w_\\co}$, where $\\co$ runs over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H, H]$. We also classify the conjugacy classes satisfying a generalization of Lusztig's conjecture \\cite{L4}.","authors_text":"Sian Nie, Xuhua He","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-12-05T03:21:51Z","title":"Minimal length elements of extended affine Coxeter groups, II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.0824","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":"sha256:837952b2f3e7d97c62a9b73f0e40512d9ecf4185ebe4eec9c3b85a946e94d38e","target":"record","created_at":"2026-05-17T23:53:19Z","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":"21504372e2457e05d7943440cc1a95d5c4a301701423535b0584c76869053039","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2011-12-05T03:21:51Z","title_canon_sha256":"d21abfd4d6bdd533c09fa666159ff5d712a5d55b5c394897117db9381d445a70"},"schema_version":"1.0","source":{"id":"1112.0824","kind":"arxiv","version":1}},"canonical_sha256":"328e78308837b8d7f7cbf7e0b8b51c4dd2eb31c0ce8462584266576cfad445c9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"328e78308837b8d7f7cbf7e0b8b51c4dd2eb31c0ce8462584266576cfad445c9","first_computed_at":"2026-05-17T23:53:19.086433Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:53:19.086433Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xKzJFcCOqcLoIZo+c3wFOZKvC+4vXLWirkdmHh6I6vWOIzb1/miI1wVf1ogt/ErL2X4un9GgXKPjDt9zf6guDA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:53:19.087055Z","signed_message":"canonical_sha256_bytes"},"source_id":"1112.0824","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:837952b2f3e7d97c62a9b73f0e40512d9ecf4185ebe4eec9c3b85a946e94d38e","sha256:79084a7244d21f8fa3c564f348e108d84483bc89565a92e3d8c8c677829e3643"],"state_sha256":"0b53ca765546fe7e2d118763c2ea895ef0c2a3c1027e377b841d03a3e531887f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mY5m/D/oBwltI+/B90KWJUJU5TKiXiUN2ZQVBakYvmB4+HpAXvp/WUscXzJYOa5whdjEDWenFttyicKYiFR0Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T01:09:53.685982Z","bundle_sha256":"0f932a05241182a9c85ffb39464f2670cc855b5393e9b93ad575a7d5b57cff72"}}