Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The fundamental invariants of 3 x 3 x 3 arrays   We determine the three fundamental invariants in the entries of a $3 \times 3
\times 3$ 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 $\mathfrak{sl}_3(\mathbb{C}) \oplus
\mathfrak{sl}_3(\mathbb{C}) \oplus \mathfrak{sl}_3(\mathbb{C})$.

28
The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
15
The categories of this paper are: nucl-ex
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio