The fundamental invariants of 3 x 3 x 3 arrays

We determine the three fundamental invariants in the entries of a $3 \times 3 \times 3$arrayover$$\mathbb{C}$$asexplicitpolynomialsinthe27variables$x_{ijk}$for$1 \le i, j, k \le 3$\mathrm{.}BytheworkofVinbergon$\theta$-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}$)$.