2011/12/13 by Murray R. Bremner, Bremner, Murray R., Jiaxiong Hu +1
Computer Science · Engineering · Mathematics · #13A50 (Primary) 15A72 #17B10 (Secondary) #Antenna Design and Optimization #Cellular Automata and Applications #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Computer and information sciences #FOS: Mathematics #Representation Theory (math.RT) #Symbolic Computation (cs.SC) #cs.SC #graph theory and CDMA systems #math.AC #math.CO #math.RT #msc:13A50 #msc:15A72 #msc:17B10
paper · pdf · doi:10.48550/arxiv.1112.2949
17 pages; revised version has more references, corrected typos, one simplified proof, and ancillary files containing the three fundamental invariants
openalex publication_date 2011/12/13 · arxiv created 2012/03/28 · arxiv updated 2015/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We determine the three fundamental invariants in the entries of a 3 × 3 × 3 array over ℂ as explicit polynomials in the 27 variables xijk for 1 ≤ i, j, k ≤ 3. By the work of Vinberg on θ-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 SL3(ℂ) × SL3(ℂ) × SL3(ℂ) acting irreducibly on its natural representation ℂ3 ⊗ ℂ3 ⊗ ℂ3. These generators have respectively 1152, 9216 and 209061 terms; we find compact expressions in terms of the orbits of the finite group (S3 × S3 × S3) \rtimes S3 acting on monomials of weight zero for the action of the Lie algebra \mathfraksl3(ℂ) ⊕ \mathfraksl3(ℂ) ⊕ \mathfraksl3(ℂ).