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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-"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1905.02180","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-05-06T17:54:35Z","cross_cats_sorted":[],"title_canon_sha256":"52bdab88e46c96c503fcf714914ed0035483076622758cfff0a43aa923de545e","abstract_canon_sha256":"e0a23fa5015acfdea9b73b613fe9349551ac71b2d7ac98653dff4748f6c85ad6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:55:57.070734Z","signature_b64":"y6aXtRfagX9rp6ZmMDlxnHt5Jv6BCbbgQLRJ+Rvt22sKqW0UZwHzuE/YMGep9yQIkr/R0F8XUVoM7P9EsSDvAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d451cc844eade96f97a1dd1eaa5b1f34c79c186e2c01d259dc648ea9c573300c","last_reissued_at":"2026-07-05T00:55:57.070191Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:55:57.070191Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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