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These automorphisms can be lifted to the affine Kac-Moody counterparts of these algebras and give a subalgebra of type $F_4^{(1)}\\oplus G_2^{(1)}$ within a type $E_8^{(1)}$ Kac-Moody Lie algebra. We will consider the level-1 irreducible $E_8^{(1)}$-module $V^{\\Lambda_0}$ and investigate its branching rule, that is how it decomposes as a direct sum of irreducible $F_4^{(1)}\\oplus G_2^{(1)}$-modules.\n  We calculate these branching rule"},"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":"2206.00163","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2022-06-01T00:37:00Z","cross_cats_sorted":[],"title_canon_sha256":"240c108a1d6e9e28c35dd591302392ab56a3d11ed82e763e406e2297a1920f6b","abstract_canon_sha256":"e29ef1e29ce9123277d10ba004bde3e303d0566bc95565946e5863d369090f60"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T01:04:29.808704Z","signature_b64":"5mCp2TnArYtoCtZ8qJzBbV/Hx2G21zJmuq2Tqq+L55r/jAZo47o12l3WC7EG+kNs+410prEgd3RZZ75j0+tUCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1297dc88bf4a735ff101da53e0e4348ede7de3dcc6062a9bda87abb41f56c89d","last_reissued_at":"2026-05-20T01:04:29.806228Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T01:04:29.806228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Branching rule decomposition of the level-1 $E_8^{(1)}$-module with respect to the irregular subalgebra $F_4^{(1)} \\oplus G_2^{(1)}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Joshua D. Carey","submitted_at":"2022-06-01T00:37:00Z","abstract_excerpt":"Given a Lie algebra of type $E_8$, one can use Dynkin diagram automorphisms of the $E_6$ and $D_4$ Dynkin diagrams to locate a subalgebra of type $F_4\\oplus G_2$. These automorphisms can be lifted to the affine Kac-Moody counterparts of these algebras and give a subalgebra of type $F_4^{(1)}\\oplus G_2^{(1)}$ within a type $E_8^{(1)}$ Kac-Moody Lie algebra. We will consider the level-1 irreducible $E_8^{(1)}$-module $V^{\\Lambda_0}$ and investigate its branching rule, that is how it decomposes as a direct sum of irreducible $F_4^{(1)}\\oplus G_2^{(1)}$-modules.\n  We calculate these branching rule"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.00163","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2206.00163/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2206.00163","created_at":"2026-05-20T01:04:29.806390+00:00"},{"alias_kind":"arxiv_version","alias_value":"2206.00163v2","created_at":"2026-05-20T01:04:29.806390+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2206.00163","created_at":"2026-05-20T01:04:29.806390+00:00"},{"alias_kind":"pith_short_12","alias_value":"CKL5ZCF7JJZV","created_at":"2026-05-20T01:04:29.806390+00:00"},{"alias_kind":"pith_short_16","alias_value":"CKL5ZCF7JJZV74IB","created_at":"2026-05-20T01:04:29.806390+00:00"},{"alias_kind":"pith_short_8","alias_value":"CKL5ZCF7","created_at":"2026-05-20T01:04:29.806390+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/CKL5ZCF7JJZV74IB3JJ6BZBUR3","json":"https://pith.science/pith/CKL5ZCF7JJZV74IB3JJ6BZBUR3.json","graph_json":"https://pith.science/api/pith-number/CKL5ZCF7JJZV74IB3JJ6BZBUR3/graph.json","events_json":"https://pith.science/api/pith-number/CKL5ZCF7JJZV74IB3JJ6BZBUR3/events.json","paper":"https://pith.science/paper/CKL5ZCF7"},"agent_actions":{"view_html":"https://pith.science/pith/CKL5ZCF7JJZV74IB3JJ6BZBUR3","download_json":"https://pith.science/pith/CKL5ZCF7JJZV74IB3JJ6BZBUR3.json","view_paper":"https://pith.science/paper/CKL5ZCF7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2206.00163&json=true","fetch_graph":"https://pith.science/api/pith-number/CKL5ZCF7JJZV74IB3JJ6BZBUR3/graph.json","fetch_events":"https://pith.science/api/pith-number/CKL5ZCF7JJZV74IB3JJ6BZBUR3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CKL5ZCF7JJZV74IB3JJ6BZBUR3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CKL5ZCF7JJZV74IB3JJ6BZBUR3/action/storage_attestation","attest_author":"https://pith.science/pith/CKL5ZCF7JJZV74IB3JJ6BZBUR3/action/author_attestation","sign_citation":"https://pith.science/pith/CKL5ZCF7JJZV74IB3JJ6BZBUR3/action/citation_signature","submit_replication":"https://pith.science/pith/CKL5ZCF7JJZV74IB3JJ6BZBUR3/action/replication_record"}},"created_at":"2026-05-20T01:04:29.806390+00:00","updated_at":"2026-05-20T01:04:29.806390+00:00"}