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It is conjectured that for each $i \\in I \\setminus \\{0\\}$ the affine Lie algebra $g$ has a positive geometric crystal whose ultra-discretization is isomorphic to the limit of certain coherent family of perfect crystals for $g^L$. We prove this conjecture for $i=2$ and $g = A_n^{(1)}$."},"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":"1209.4565","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2012-09-20T15:24:52Z","cross_cats_sorted":[],"title_canon_sha256":"9f5cbc883b79e4b00efbc28c58b00ae68c5aec91f813c7cc3cbe687f6cfba35c","abstract_canon_sha256":"d818d062c46b8fc18abfa76b16ba5d4d596af388e06fe53f06cd22fe93b755e8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:11.306473Z","signature_b64":"T+LpzGbGKLdg22cvl5l7Y3BgQ8ZRHkULAKYq/UwhIKtH6Po508JLz8LKyYHEx5E8rGezFWFKHohONsxzOM0iDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a143579ed4a0067d9f8d7e0eff7b223f08c9d988a449bf219e670a1f9237a7dc","last_reissued_at":"2026-05-18T03:45:11.305701Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:11.305701Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$A_n^{(1)}$-Geometric Crystal corresponding to Dynkin index $i=2$ and its ultra-discretization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.QA","authors_text":"Kailash C. 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