Fundamental invariants for the action of $SL_3(\mathbb {C}) \times SL_3(\mathbb {C}) \times SL_3(\mathbb {C})$ on $3 \times 3 \times 3$ 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 $SL_3(\mathbb {C}) \times SL_3(\mathbb {C}) \times SL_3(\mathbb {C})$ 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 $\mathfrak {sl}_3(\mathbb {C}) \oplus \mathfrak {sl}_3(\mathbb {C}) \oplus \mathfrak {sl}_3(\mathbb {C})$.