The 3 ×  3 ×  3 Hyperdeterminant as a Polynomial in the Fundamental Invariants for $${{SL_3(\mathbb{C}) \times SL_3(\mathbb{C}) \times SL_3(\mathbb{C})}}$$ S L 3 ( C ) × S L 3 ( C ) × S L 3 ( C )

Genuine tripartite entanglement as a probe of quantum phase transitions in a spin-1 Heisenberg chain with single-ion anisotropy

“…A direct computation yields Q α , Q β , E α , E β and the fundamental invariant I 9 for Nurmiev's normal form [59]…”

“…According to Theorem 3.1 of [59], the 3 × 3 × 3 hyperdeterminant ∆ 333 (homogeneous polynomial of degree 36) for a general trilinear form can be generated from the three fundamental SL(3, C) ⊗3 invariants I 6 , I 9 and J 12 as where e i j ≡ η i ζ j , e (i j) ≡ e i j + e ji , e (i j) ≡ e i j + ae ji , f i j ≡ ξ i ζ j , f (i j) ≡ f i j + f ji , π i ≡ ∂ f 0 /∂x i , and Π i ≡ ∂ f 0 /∂y i . Using Eqs.…”

“…Nevertheless, one may still express the three-qutrit hyperdeterminant in terms of the three fundamental SL(3, C) ⊗3 invariants I 6 , I 9 and I 12 , which are homogeneous polynomials of degrees 6, 9 and 12 respectively. The relation between the three-qutrit hyperdeterminant ∆ 333 and the three fundamental invariants I 6 , I 9 and I 12 is given by [59]…”

“…A direct computation shows that Bremner's invariant (denoted as J 12 ) and Briand's invariant (denoted as I 12 ) satisfy the simple relation −I 12 − I 2 6 = 24J 12 . Hence, Bremner's invariant J 12 for Nurmiev's normal form can be explicitly written as[59]…”

Fundamental invariants of tensors, Latin hypercubes, and rectangular Kronecker coefficients

“…(To compare with generators in [BH13], ∆ T 1 , ∆ T 2 differ up to scalar, and up to ∆ T 3 + c∆ 2 T 1 .) Note that the geometric hyperdeterminant of format 3×3×3 has degree 36 and can be written as P (∆ T 1 , ∆ T 2 , ∆ T 3 ) for some polynomial P , see [BHO14] for an explicit presentation. In expansion of P , each monomial of type ∆ α T 1 ∆ β T 2 ∆ γ T 3 can be written as a 3 × 36 balanced table: horizontally concatenate T 1 α times, T 2 β times and T 3 γ times, so that each concatenated table is shifted in numbers (see § 3.5 for this operation).…”