The fundamental invariants of 3 x 3 x 3 arrays

TL;DR

This paper explicitly constructs the three fundamental polynomial invariants of a 3x3x3 array under a specific group action, revealing their structure and symmetries.

Contribution

It provides explicit polynomial formulas for the invariants of 3x3x3 arrays under the action of SL(3,C)^3, expanding understanding of their algebraic structure.

Findings

Invariants have degrees 6, 9, and 12.

Explicit polynomial expressions are derived.

Invariants are expressed via group orbit sums.

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 $SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})$ acting irreducibly on its natural representation $\mathbb{C}^3 \otimes \mathbb{C}^3 \otimes \mathbb{C}^3$. These generators have respectively 1152, 9216 and 209061 terms; we find compact expressions in terms of the orbits of the finite group $(S_3 \times S_3 \times S_3) \rtimes S_3$ acting on monomials of weight zero for the action of the Lie algebra $\mathfrak{sl}_3(\mathbb{C}) \oplus \mathfrak{sl}_3(\mathbb{C}) \oplus \mathfrak{sl}_3(\mathbb{C})$.