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We develop combinatorial methods to reveal refined structural properties of such objects.\n  Firstly, we study subcrystals of a regular $A_n$-crystal $K$ and characterize pairwise intersections of maximal subcrystals with colors $1,...,n-1$ and colors $2,...,n$. This leads to a recursive description of the structure of $K$ and pro"},"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":"1201.4549","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-22T11:04:27Z","cross_cats_sorted":[],"title_canon_sha256":"3a5c4c6531d5c28f63ba64c3204bdf1e4976644053618f9f30eafd6add3afaa4","abstract_canon_sha256":"c82ff76c900eb0b41028820e28dd7643d297f220b82879114eab0cbbce8adc92"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:48:44.984039Z","signature_b64":"gYthXEFX2EIp66BKW09WwOvWrF+I4ZZGLOnXsfRB/louoU4NI2yI4cir7aY35aryA4VQ1Quc1HNmFtgRbJY/AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e03a19397365224225c641ad55d2991013ba398853f3a855ac2eaa5151718d54","last_reissued_at":"2026-05-18T03:48:44.983375Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:48:44.983375Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the combinatorial structure of crystals of types A,B,C","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Karzanov, Gleb Koshevoy, Vladimir Danilov","submitted_at":"2012-01-22T11:04:27Z","abstract_excerpt":"Regular $A_n$-, $B_n$- and $C_n$-crystals are edge-colored directed graphs, with ordered colors $1,2,...,n$, which are related to representations of quantized algebras $U_q(\\mathfrak{sl}_{n+1})$, $U_q(\\mathfrak{sp}_{2n})$ and $U_q(\\mathfrak{so}_{2n+1})$, respectively. We develop combinatorial methods to reveal refined structural properties of such objects.\n  Firstly, we study subcrystals of a regular $A_n$-crystal $K$ and characterize pairwise intersections of maximal subcrystals with colors $1,...,n-1$ and colors $2,...,n$. 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