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Geiss, Leclerc and Schr$\\ddot{\\rm o}$er conjectured that $\\mathcal {T}_\\Lambda$ is connected, see [C.Geiss, B.Leclerc, J.Schr\\\"{o}er, Rigid modules over preprojective algebras, Invent.Math., 165(2006), 589-632]. In this paper, we prove that this conjecture is true when $\\Lambda$ is of representation finite type or tame type. Moreover, we also prove that $\\mathcal {T}_\\La"},"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":"1301.3983","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-01-17T05:04:35Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"aedd1e2d9eac8588ea54fa159ac6248c627ccd760f0154c3e98c00dfe800cbef","abstract_canon_sha256":"50b26b4fdd16a8ebf0745062cf0e240b930ac515d13c8ebda7557a87f5047674"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:36:14.718633Z","signature_b64":"SRDehjazM7hJJXoyPuTEdSYz5xd1kayqCvmfjlnt9M+q4Ek/vTeM+GIko6tp5sHgoosfq3Z6N/jiLCySQ55jCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c794b5d240bbf93a3249096e4e86b1ed27445251ec1d3f896250403b0a8dfaf","last_reissued_at":"2026-05-18T03:36:14.718126Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:36:14.718126Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mutation graphs of maximal rigid modules over finite dimensional preprojective algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.RT","authors_text":"Hongbo Yin, Shunhua Zhang","submitted_at":"2013-01-17T05:04:35Z","abstract_excerpt":"Let $Q$ be a finite quiver of Dynkin type and $\\Lambda=\\Lambda_Q$ be the preprojective algebra of $Q$ over an algebraically closed field $k$. Let $\\mathcal {T}_\\Lambda$ be the mutation graph of maximal rigid $\\Lambda$ modules. Geiss, Leclerc and Schr$\\ddot{\\rm o}$er conjectured that $\\mathcal {T}_\\Lambda$ is connected, see [C.Geiss, B.Leclerc, J.Schr\\\"{o}er, Rigid modules over preprojective algebras, Invent.Math., 165(2006), 589-632]. In this paper, we prove that this conjecture is true when $\\Lambda$ is of representation finite type or tame type. Moreover, we also prove that $\\mathcal {T}_\\La"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.3983","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1301.3983","created_at":"2026-05-18T03:36:14.718197+00:00"},{"alias_kind":"arxiv_version","alias_value":"1301.3983v1","created_at":"2026-05-18T03:36:14.718197+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.3983","created_at":"2026-05-18T03:36:14.718197+00:00"},{"alias_kind":"pith_short_12","alias_value":"TR4UWXJEBO7Z","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_16","alias_value":"TR4UWXJEBO7ZHIZE","created_at":"2026-05-18T12:28:02.375192+00:00"},{"alias_kind":"pith_short_8","alias_value":"TR4UWXJE","created_at":"2026-05-18T12:28:02.375192+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TR4UWXJEBO7ZHIZESCLOJ2DLD3","json":"https://pith.science/pith/TR4UWXJEBO7ZHIZESCLOJ2DLD3.json","graph_json":"https://pith.science/api/pith-number/TR4UWXJEBO7ZHIZESCLOJ2DLD3/graph.json","events_json":"https://pith.science/api/pith-number/TR4UWXJEBO7ZHIZESCLOJ2DLD3/events.json","paper":"https://pith.science/paper/TR4UWXJE"},"agent_actions":{"view_html":"https://pith.science/pith/TR4UWXJEBO7ZHIZESCLOJ2DLD3","download_json":"https://pith.science/pith/TR4UWXJEBO7ZHIZESCLOJ2DLD3.json","view_paper":"https://pith.science/paper/TR4UWXJE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1301.3983&json=true","fetch_graph":"https://pith.science/api/pith-number/TR4UWXJEBO7ZHIZESCLOJ2DLD3/graph.json","fetch_events":"https://pith.science/api/pith-number/TR4UWXJEBO7ZHIZESCLOJ2DLD3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TR4UWXJEBO7ZHIZESCLOJ2DLD3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TR4UWXJEBO7ZHIZESCLOJ2DLD3/action/storage_attestation","attest_author":"https://pith.science/pith/TR4UWXJEBO7ZHIZESCLOJ2DLD3/action/author_attestation","sign_citation":"https://pith.science/pith/TR4UWXJEBO7ZHIZESCLOJ2DLD3/action/citation_signature","submit_replication":"https://pith.science/pith/TR4UWXJEBO7ZHIZESCLOJ2DLD3/action/replication_record"}},"created_at":"2026-05-18T03:36:14.718197+00:00","updated_at":"2026-05-18T03:36:14.718197+00:00"}