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We find a simple algorithm which describes the set $\\Psi(\\lambda)$ of all $\\mathtt{I}$-orbits in $\\overline{\\mathtt{X}}_\\lambda$ in terms of coweights. We introduce $R$-operators (associated to positive roots) on the coweight lattice of $G$, which exactly describe the closure relation of $\\mathtt{I}$-orbits. These operators satisfy Bra"},"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":"1906.09341","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-06-21T21:49:30Z","cross_cats_sorted":["math.AG","math.CO","math.GR"],"title_canon_sha256":"e6ace376ff37263076b30bbbeb2a399294c24f534d51e5acdc69c328558effc0","abstract_canon_sha256":"bf7a29fec7392759169479bb447f5105ce021f1e1e52bb751d6c8a0118a9b6b6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:38:45.564789Z","signature_b64":"msTa6UmCE1mZA+Lb26NPl5KiAIjBoPNLSWpCuhmezpBY76ITQP3DQFqbhqAQPstST1mI1Wu8FKTpCxPHBFoQBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"987c285e1b3c161399d971b74ab3e2e39b35bdb4ed9c58d44521506c1f5cc3b0","last_reissued_at":"2026-07-05T00:38:45.564332Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:38:45.564332Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A combinatorial study of affine Schubert varieties in affine Grassmannian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CO","math.GR"],"primary_cat":"math.RT","authors_text":"Jiuzu Hong, Marc Besson","submitted_at":"2019-06-21T21:49:30Z","abstract_excerpt":"Let $\\overline{\\mathtt{X}}_\\lambda$ be the closure of the $\\mathtt{I}$-orbit $\\mathtt{X}_\\lambda$ in the affine Grassmanian $\\mathtt{Gr}$ of a simple algebraic group $G$ of adjoint type, where $\\mathtt{I}$ is the Iwahori group and $\\lambda$ is a coweight of $G$. We find a simple algorithm which describes the set $\\Psi(\\lambda)$ of all $\\mathtt{I}$-orbits in $\\overline{\\mathtt{X}}_\\lambda$ in terms of coweights. We introduce $R$-operators (associated to positive roots) on the coweight lattice of $G$, which exactly describe the closure relation of $\\mathtt{I}$-orbits. 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