{"paper":{"title":"Shard theory for $g$-fans","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Yuya Mizuno","submitted_at":"2022-12-21T03:29:28Z","abstract_excerpt":"For a finite dimensional algebra $A$, the notion of $g$-fan $\\Sigma(A)$ is defined from two-term silting complexes of $A$ in the real Grothendieck group $K_0(\\mathsf{proj} A)_{\\mathbb{R}}$. In this paper, we discuss the theory of shards to $\\Sigma(A)$, which was originally defined for a hyperplane arrangement. We establish a correspondence between the set of join-irreducible elements of the poset of torsion classes of $\\mathrm{mod} A$ and the set of shards of $\\Sigma(A)$ for $g$-finite algebra $A$. Moreover, we show that the semistable region of a brick of $\\mathrm{mod} A$ is exactly given by "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2212.10745","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/2212.10745/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"}