{"paper":{"title":"Extended weak order for the affine symmetric group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Grant T. Barkley","submitted_at":"2025-02-09T12:24:03Z","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 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.05875","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.05875/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}