The fundamental invariants of 3 x 3 x 3 arrays

Abstract

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 $S{L}_3(\mathbb{C})\times S{L}_3(\mathbb{C})\times S{L}_3(\mathbb{C})$ acting irreducibly on its natural representation ${\mathbb{C}}^3\otimes {\mathbb{C}}^3\otimes {\mathbb{C}}^3$. These generators have respectively 1152, 9216 and 209061 terms; we find compact expressions in terms of the orbits of the finite group $(S_3\times S_3\times S_3)⋊S_3$ acting on monomials of weight zero for the action of the Lie algebra ${\mathfrak{sl}}_3(\mathbb{C})\oplus {\mathfrak{sl}}_3(\mathbb{C})\oplus {\mathfrak{sl}}_3(\mathbb{C})$.

arXiv comment17 pages; revised version has more references, corrected typos, one simplified proof, and ancillary files containing the three fundamental invariants

References in corpus (2)

• Symplectic bundles on the plane, secant varieties and Lüroth quartics revisited
• Defining relation for semi-invariants of three by three matrix triples

Cited by in corpus (1)

• Noncommutative resolutions and CICY quotients from a non-abelian GLSM