{"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 $\\mathbb{C}^3 \\otimes \\mathbb{C}^3 \\otimes \\mathbb{C}^3$. These generators have respectively 1152, 9216 and 20"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.2949","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":""},"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"}