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We construct the integrable crystals $\\mathbf{B}^{G}(\\lambda),\\ \\lambda\\in\\Lambda^{+}_{G}$, using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group. We construct the tensor product maps $\\mathbf{p}_{\\lambda_{1},\\lambda_{2}}\\colon \\mathbf{B}^{G}(\\lambda_{1}) \\otimes \\mathbf{B}^{G}(\\lambda_{2}) \\rightarrow \\mathbf{B}^{G}(\\lambda_{1}+\\lambda_{2})\\cup\\{0\\}$ in terms of multiplication of generalized transversal slices"},"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":"1709.00391","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-09-01T16:53:03Z","cross_cats_sorted":[],"title_canon_sha256":"7c8b79d716a6580d38e7b345669b52664ec98b110a5a1908e1ddfc03c3ce6048","abstract_canon_sha256":"eb145c867ea4db3b56f425d5a09118c379792924dc54ee28ca697f95b0ed9b86"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:19:03.718427Z","signature_b64":"gv6nb2HOEhqrsljnsCZDozA8ApB4t3yE3xVL8PVM5S3Usgij0BJxhZ2bIdfBxwDIbO6Z09Dcs00dl+0f/KpaDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0097545dcbeabe42568fd975c85b98f0385185111b1def0a09b5acbc28ecad8","last_reissued_at":"2026-05-18T00:19:03.717673Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:19:03.717673Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Integrable crystals and restriction to Levi via generalized slices in the affine Grassmannian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Vasily Krylov","submitted_at":"2017-09-01T16:53:03Z","abstract_excerpt":"Let $G$ be a connected reductive algebraic group over $\\mathbb{C}$. Let $\\Lambda^{+}_{G}$ be the monoid of dominant weights of $G$. We construct the integrable crystals $\\mathbf{B}^{G}(\\lambda),\\ \\lambda\\in\\Lambda^{+}_{G}$, using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group. We construct the tensor product maps $\\mathbf{p}_{\\lambda_{1},\\lambda_{2}}\\colon \\mathbf{B}^{G}(\\lambda_{1}) \\otimes \\mathbf{B}^{G}(\\lambda_{2}) \\rightarrow \\mathbf{B}^{G}(\\lambda_{1}+\\lambda_{2})\\cup\\{0\\}$ in terms of multiplication of generalized transversal slices"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.00391","kind":"arxiv","version":3},"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":"1709.00391","created_at":"2026-05-18T00:19:03.717797+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.00391v3","created_at":"2026-05-18T00:19:03.717797+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.00391","created_at":"2026-05-18T00:19:03.717797+00:00"},{"alias_kind":"pith_short_12","alias_value":"6AEXKRO4X2V6","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_16","alias_value":"6AEXKRO4X2V6IJLI","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_8","alias_value":"6AEXKRO4","created_at":"2026-05-18T12:31:03.183658+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2201.08386","citing_title":"A mathematical definition of Coulomb branches of supersymmetric gauge theories and geometric Satake correspondences for Kac-Moody Lie algebras","ref_index":19,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6AEXKRO4X2V6IJLI7WLVZBNZR4","json":"https://pith.science/pith/6AEXKRO4X2V6IJLI7WLVZBNZR4.json","graph_json":"https://pith.science/api/pith-number/6AEXKRO4X2V6IJLI7WLVZBNZR4/graph.json","events_json":"https://pith.science/api/pith-number/6AEXKRO4X2V6IJLI7WLVZBNZR4/events.json","paper":"https://pith.science/paper/6AEXKRO4"},"agent_actions":{"view_html":"https://pith.science/pith/6AEXKRO4X2V6IJLI7WLVZBNZR4","download_json":"https://pith.science/pith/6AEXKRO4X2V6IJLI7WLVZBNZR4.json","view_paper":"https://pith.science/paper/6AEXKRO4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.00391&json=true","fetch_graph":"https://pith.science/api/pith-number/6AEXKRO4X2V6IJLI7WLVZBNZR4/graph.json","fetch_events":"https://pith.science/api/pith-number/6AEXKRO4X2V6IJLI7WLVZBNZR4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6AEXKRO4X2V6IJLI7WLVZBNZR4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6AEXKRO4X2V6IJLI7WLVZBNZR4/action/storage_attestation","attest_author":"https://pith.science/pith/6AEXKRO4X2V6IJLI7WLVZBNZR4/action/author_attestation","sign_citation":"https://pith.science/pith/6AEXKRO4X2V6IJLI7WLVZBNZR4/action/citation_signature","submit_replication":"https://pith.science/pith/6AEXKRO4X2V6IJLI7WLVZBNZR4/action/replication_record"}},"created_at":"2026-05-18T00:19:03.717797+00:00","updated_at":"2026-05-18T00:19:03.717797+00:00"}