{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:RYP62CPM3LEZSZDUFHJCIG4T63","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":"37c1d88be63cfd36544d20d9302a73e5eb89b4bc4618193e23c05eb9a6a8c1cd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-09T12:24:03Z","title_canon_sha256":"69c56c880753b4f866bf955587a214ac3f8847bb1613c2f6c48cc183539ccfab"},"schema_version":"1.0","source":{"id":"2502.05875","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2502.05875","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"arxiv_version","alias_value":"2502.05875v1","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.05875","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"pith_short_12","alias_value":"RYP62CPM3LEZ","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"pith_short_16","alias_value":"RYP62CPM3LEZSZDU","created_at":"2026-07-05T10:11:44Z"},{"alias_kind":"pith_short_8","alias_value":"RYP62CPM","created_at":"2026-07-05T10:11:44Z"}],"graph_snapshots":[{"event_id":"sha256:98d1e253799b622cf7a8f03bc505b497d5a2393033c804e4b8f0f5e20612dc60","target":"graph","created_at":"2026-07-05T10:11:44Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2502.05875/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The extended weak order on a Coxeter group $W$ is the poset of biclosed sets in its root system. In (Barkley-Speyer 2024), it was shown that when $W=\\widetilde{S}_n$ is the affine symmetric group, then the extended weak order is a quotient of the lattice $L_n$ of translation-invariant total orderings of the integers. In this article, we give a combinatorial introduction to $L_n$ and the extended weak order on $\\widetilde{S}_n$. We show that $L_n$ is an algebraic completely semidistributive lattice. We describe its canonical join representations using a cyclic version of Reading's non-crossing ","authors_text":"Grant T. Barkley","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-09T12:24:03Z","title":"Extended weak order for the affine symmetric group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.05875","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:d5fccb82dbc7bea51142da3e3a945553cb1c03fada18bf23a2fc426c0af157f8","target":"record","created_at":"2026-07-05T10:11:44Z","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":"37c1d88be63cfd36544d20d9302a73e5eb89b4bc4618193e23c05eb9a6a8c1cd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-02-09T12:24:03Z","title_canon_sha256":"69c56c880753b4f866bf955587a214ac3f8847bb1613c2f6c48cc183539ccfab"},"schema_version":"1.0","source":{"id":"2502.05875","kind":"arxiv","version":1}},"canonical_sha256":"8e1fed09ecdac999647429d2241b93f6c12d051277cfb39a0444825fccd81efc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8e1fed09ecdac999647429d2241b93f6c12d051277cfb39a0444825fccd81efc","first_computed_at":"2026-07-05T10:11:44.560924Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:11:44.560924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tMfvd8HrrcJKrkFsgtnNfSwt/BiK9m+UXwS8vBXBOhPnJsJxf44IvjCxsY0OANQaOL2N3S5JXXexM1Qk+BBdAw==","signature_status":"signed_v1","signed_at":"2026-07-05T10:11:44.561416Z","signed_message":"canonical_sha256_bytes"},"source_id":"2502.05875","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d5fccb82dbc7bea51142da3e3a945553cb1c03fada18bf23a2fc426c0af157f8","sha256:98d1e253799b622cf7a8f03bc505b497d5a2393033c804e4b8f0f5e20612dc60"],"state_sha256":"9778d9290782c2e1e5d52da82a6d231181d7438e874bc2e4fefc676eae303a70"}