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By the work of Vinberg on $\\theta$-groups, it is known that these homogeneous polynomials have degrees 6, 9 and 12; they freely generate the algebra of invariants for the Lie group $SL_3(\\mathbb{C}) \\times SL_3(\\mathbb{C}) \\times SL_3(\\mathbb{C})$ acting irreducibly on its natural representation $\\mathbb{C}^3 \\otimes \\mathbb{C}^3 \\otimes \\mathbb{C}^3$. These generators have respectively 1152, 9216 and 20"},"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":"1112.2949","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-12-13T16:40:21Z","cross_cats_sorted":["cs.SC","math.CO","math.RT"],"title_canon_sha256":"899e49cfc9ee42c5f88c223c07de197b01e5ad2ece7436dab2be28122470c157","abstract_canon_sha256":"b112e5255da51f368f5523e3771e1646ad392dee7d4d2af51fb01d8e53bb4d6e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:21:27.238700Z","signature_b64":"gXSWUhSgg9q3mMjJEGpy6mFyva1H617FefF3SdZ+x/LyDz10Pw7XduQfU8hOTnjxCZiCHnb9s05gwHVl/pkQCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a1f8018093f9313d8cd5ebc1103ed915e107dd7669577c38dd947a9c66670ee3","last_reissued_at":"2026-05-18T02:21:27.237974Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:21:27.237974Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The fundamental invariants of 3 x 3 x 3 arrays","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SC","math.CO","math.RT"],"primary_cat":"math.AC","authors_text":"Jiaxiong Hu, Murray R. Bremner","submitted_at":"2011-12-13T16:40:21Z","abstract_excerpt":"We determine the three fundamental invariants in the entries of a $3 \\times 3 \\times 3$ array over $\\mathbb{C}$ as explicit polynomials in the 27 variables $x_{ijk}$ for $1 \\le i, j, k \\le 3$. By the work of Vinberg on $\\theta$-groups, it is known that these homogeneous polynomials have degrees 6, 9 and 12; they freely generate the algebra of invariants for the Lie group $SL_3(\\mathbb{C}) \\times SL_3(\\mathbb{C}) \\times SL_3(\\mathbb{C})$ acting irreducibly on its natural representation $\\mathbb{C}^3 \\otimes \\mathbb{C}^3 \\otimes \\mathbb{C}^3$. 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