{"paper":{"title":"The wall-chamber structures of the real Grothendieck groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Sota Asai","submitted_at":"2019-05-06T17:54:35Z","abstract_excerpt":"For a finite-dimensional algebra $A$ over a field $K$ with $n$ simple modules, the real Grothendieck group $K_0(\\operatorname{\\mathsf{proj}} A)_\\mathbb{R}:=K_0(\\operatorname{\\mathsf{proj}} A) \\otimes_\\mathbb{Z} \\mathbb{R} \\cong \\mathbb{R}^n$ gives stability conditions of King. We study the associated wall-chamber structure of $K_0(\\operatorname{\\mathsf{proj}} A)_\\mathbb{R}$ by using the Koenig--Yang correspondences in silting theory. First, we introduce an equivalence relation on $K_0(\\operatorname{\\mathsf{proj}} A)_\\mathbb{R}$ called TF equivalence by using numerical torsion pairs of Baumann-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.02180","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1905.02180/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"}