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For $\\lambda \\in P_+$, let $L(\\lambda)$ be the irreducible, integrable, highest weight representation of $\\fg$ with highest weight $\\lambda$. For a positive integer $s$, define the {\\em saturated tensor semigroup} as \\begin{align*} \\Gamma_s:= \\{(\\lambda_1, \\dots, \\lambda_s,\\mu)\\in P_+^{s+1}: \\exists\\, N>1 \\,\\,\\text{with}\\,\\, L(N\\mu)\\subset L(N\\lambda_1)\\otimes \\dots \\otimes L(N\\lambda_s)\\}. \\end{align*} The aim of this paper is "},"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":"1306.0073","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2013-06-01T04:05:05Z","cross_cats_sorted":[],"title_canon_sha256":"e915f473d2d92896f786e347b658713333a14c55d4a4b0c692f34afa2569d7b3","abstract_canon_sha256":"7d66ee5a149add847567d451f94391aa79113daacf36d47f4e9ac8c8283d9536"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:56.223896Z","signature_b64":"xiCfbgkuO8Cda9moVbOfHHOtyBIW3h5ny6K61twF+37Qn/9H0Dv1Y/IdtuB9q1qgnnaKsMsNATGRS2JTF06+Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c3ca7e1142e67a843cccad29bbb48397c89f423f58ef55b3a511415e2531dd12","last_reissued_at":"2026-05-18T03:21:56.223509Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:56.223509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A study of saturated tensor cone for symmetrizable Kac-Moody algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Merrick Brown, Shrawan Kumar","submitted_at":"2013-06-01T04:05:05Z","abstract_excerpt":"Let $\\fg$ be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra $\\fh$ and the Weyl group $W$. Let $P_+$ be the set of dominant integral weights. For $\\lambda \\in P_+$, let $L(\\lambda)$ be the irreducible, integrable, highest weight representation of $\\fg$ with highest weight $\\lambda$. For a positive integer $s$, define the {\\em saturated tensor semigroup} as \\begin{align*} \\Gamma_s:= \\{(\\lambda_1, \\dots, \\lambda_s,\\mu)\\in P_+^{s+1}: \\exists\\, N>1 \\,\\,\\text{with}\\,\\, L(N\\mu)\\subset L(N\\lambda_1)\\otimes \\dots \\otimes L(N\\lambda_s)\\}. \\end{align*} The aim of this paper is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0073","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":"1306.0073","created_at":"2026-05-18T03:21:56.223561+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.0073v1","created_at":"2026-05-18T03:21:56.223561+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.0073","created_at":"2026-05-18T03:21:56.223561+00:00"},{"alias_kind":"pith_short_12","alias_value":"YPFH4EKC4Z5I","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_16","alias_value":"YPFH4EKC4Z5IIPGM","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_8","alias_value":"YPFH4EKC","created_at":"2026-05-18T12:28:09.283467+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/YPFH4EKC4Z5IIPGMVUU3XNEDS7","json":"https://pith.science/pith/YPFH4EKC4Z5IIPGMVUU3XNEDS7.json","graph_json":"https://pith.science/api/pith-number/YPFH4EKC4Z5IIPGMVUU3XNEDS7/graph.json","events_json":"https://pith.science/api/pith-number/YPFH4EKC4Z5IIPGMVUU3XNEDS7/events.json","paper":"https://pith.science/paper/YPFH4EKC"},"agent_actions":{"view_html":"https://pith.science/pith/YPFH4EKC4Z5IIPGMVUU3XNEDS7","download_json":"https://pith.science/pith/YPFH4EKC4Z5IIPGMVUU3XNEDS7.json","view_paper":"https://pith.science/paper/YPFH4EKC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.0073&json=true","fetch_graph":"https://pith.science/api/pith-number/YPFH4EKC4Z5IIPGMVUU3XNEDS7/graph.json","fetch_events":"https://pith.science/api/pith-number/YPFH4EKC4Z5IIPGMVUU3XNEDS7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YPFH4EKC4Z5IIPGMVUU3XNEDS7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YPFH4EKC4Z5IIPGMVUU3XNEDS7/action/storage_attestation","attest_author":"https://pith.science/pith/YPFH4EKC4Z5IIPGMVUU3XNEDS7/action/author_attestation","sign_citation":"https://pith.science/pith/YPFH4EKC4Z5IIPGMVUU3XNEDS7/action/citation_signature","submit_replication":"https://pith.science/pith/YPFH4EKC4Z5IIPGMVUU3XNEDS7/action/replication_record"}},"created_at":"2026-05-18T03:21:56.223561+00:00","updated_at":"2026-05-18T03:21:56.223561+00:00"}