Cayley Hyperdeterminant Essentials

Updated 2 February 2026

Cayley Hyperdeterminant is a canonical generalization of matrix determinants to multidimensional arrays, uniquely characterized as a quartic invariant under SL₂³ action.
It underpins invariant theory and representation theory while providing tools for analyzing tensor symmetries and multipartite entanglement in quantum information.
Explicit formulations, quantum deformations, and combinatorial generalizations enable practical computation and extend its applications to algebraic geometry and high-dimensional optimization.

Cayley’s hyperdeterminant is the canonical generalization of the determinant from matrices to multidimensional arrays (“hypermatrices”), with foundational roles in invariant theory, representation theory, combinatorics, quantum information, and algebraic geometry. For the $2 \times 2 \times 2$ case, it is a quartic polynomial in eight variables—the minimal nonconstant SL-invariant—and serves as a template for invariants associated to higher-order and higher-dimensional tensors. Extensive structural results characterize its algebraic uniqueness, symmetries, vanishing criteria, and applications spanning from combinatorial representation-theoretic enumeration to quantum entanglement and beyond.

1. Algebraic Definition and Invariant-Theoretic Characterization

Let $A = (a_{ijk})$ , %%%%2%%%% be a $2 \times 2 \times 2$ tensor with entries viewed as coordinate functions on $V = \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$ . The group $SL_2(\mathbb{C})^3$ acts naturally on $V$ and the polynomial ring $P = \mathbb{C}[a_{000}, \ldots, a_{111}]$ . Cayley’s hyperdeterminant, denoted $\mathrm{Det}(A)$ , is the simplest nonconstant polynomial invariant under $SL_2(\mathbb{C})^3$ .

Classical invariant theory shows:

Any invariant homogeneous polynomial in the $a_{ijk}$ must have degree divisible by 4.
The space of degree-4 $SL_2(\mathbb{C})^3$ -invariants is one-dimensional, spanned by a unique (up to scaling) quartic, which is Cayley’s hyperdeterminant (Bremner et al., 2011).

The explicit quartic form for the $2 \times 2 \times 2$ hyperdeterminant is

$\begin{aligned} \mathrm{Det}(A) =\; & a_{000}^2 a_{111}^2 + a_{001}^2 a_{110}^2 + a_{010}^2 a_{101}^2 + a_{011}^2 a_{100}^2 \ & - 2(a_{000}a_{001}a_{110}a_{111} + a_{000}a_{010}a_{101}a_{111} + a_{000}a_{011}a_{100}a_{111}\ & \qquad + a_{001}a_{010}a_{101}a_{110} + a_{001}a_{011}a_{100}a_{110} + a_{010}a_{011}a_{100}a_{101}) \ & + 4(a_{000}a_{011}a_{101}a_{110} + a_{001}a_{010}a_{100}a_{111}). \end{aligned}$

Invariance is ensured by verifying annihilation under all $SL_2$ Lie algebra raising operators and Cartan elements acting in all three directions, confirming full $SL_2^3$ invariance (Bremner et al., 2011, Gibbs, 2010, Holweck et al., 2023).

2. Representation Theory and Combinatorial Structure

The invariant theory for $2 \times 2 \times 2$ arrays is addressed via decomposition into weight spaces. Given the natural $SL_2^3$ -action, the space $P_4$ of degree-4 polynomials decomposes as a direct sum of irreducibles, and the invariant subspace corresponds to the multiplicity of the trivial module $V(0) \otimes V(0) \otimes V(0)$ .

A monomial $m$ in the $a_{ijk}$ has weight $(\alpha,\beta,\gamma)$ . The zero-weight subspace is determined by the balance of exponents across parallel slices. There are exactly 12 nonnegative integer exponent arrays summing to 4 that satisfy the three slice-balance constraints, forming a basis for the 12-dimensional zero-weight space (Bremner et al., 2011).

The kernel of a linear map induced by the raising operators within this space is 1-dimensional; thus, the hyperdeterminant is unique in degree 4. Higher powers generate all polynomial invariants: the $d$ -degree invariants ( $d$ divisible by $4$) are spanned by $\mathrm{Det}(A)^{d/4}$ , with no invariants in other degrees. This yields the isomorphism $\mathbb{C}[a_{ijk}]^{SL_2^3} \cong \mathbb{C}[\mathrm{Det}(A)]$ (Bremner et al., 2011).

This weight- and slice-based combinatorial approach generalizes to arbitrary $n_1 \times \cdots \times n_k$ formats under $SL_{n_1} \times \cdots \times SL_{n_k}$ ; the minimal-degree invariant—when it exists—is always a unique hyperdeterminant up to scaling.

3. Explicit and Generalized Formulations

The most general version of Cayley’s first (combinatorial) hyperdeterminant for an order- $d$ $n \times \cdots \times n$ tensor $T$ over a field $F$ is

$\mathrm{Det}(T) = \frac{1}{n!} \sum_{\sigma_1,\ldots,\sigma_d \in S_n} \left( \prod_{k=1}^d \operatorname{sgn}(\sigma_k) \right) \prod_{i=1}^n T( \sigma_1(i), \ldots, \sigma_d(i) ),$

where $S_n$ is the permutation group on $n$ symbols (Amanov et al., 2021, Dobes, 30 Nov 2025, Jing et al., 2014). For odd $d$ , this invariant vanishes identically; for even $d=2m$ , it reduces to the standard determinant when $m=1$ .

This hyperdeterminant exhibits:

Full skew-symmetry in all directions.
$\prod_{k=1}^d \mathrm{GL}_n(F)$ -relative invariance for even $d$ :

$\mathrm{Det}\big( (A_1, \ldots, A_d) \cdot T \big) = \left( \prod_{k=1}^d \det A_k \right) \mathrm{Det}(T).$

(Amanov et al., 2021, Jing et al., 2014).

A Levi-Civita tensor contraction formulation and a Kronecker-product-based flattening enable efficient computation in the symmetric case, with recent algorithms yielding $O(N^{d(d-1)})$ complexity for order- $N$ , side-length- $d$ symmetric hypermatrices, given precomputation of appropriate elimination and duplication matrices (Dobes, 30 Nov 2025).

4. Quantum and Combinatorial Deformations

Quantum generalizations (taking $q \ne 1$ ) deform the classical hyperdeterminant to noncommuting settings. The $q$ -hyperdeterminant for an even-dimensional $n \times \ldots \times n$ array $A$ is

$\operatorname{Det}_q(A) = [n]_{q^2}! \sum_{\sigma_1, \ldots, \sigma_{2m} \in S_n} (-q)^{\sum_{k=1}^{2m} \ell(\sigma_k)} \prod_{i=1}^n a_{\sigma_1(i), \ldots, \sigma_{2m}(i)},$

where $[n]_{q^2}!$ is the quantum factorial and $\ell(\sigma)$ is the inversion number (Jing et al., 2014). The $q$ -hyperdeterminant enjoys comodule-algebra structure under quantum groups and satisfies quantum Laplace-expansion and Plücker-type relations. The $q \to 1$ limit recovers the classical Cayley hyperdeterminant.

Combinatorial deformations using alternating sign matrices (ASMs) further generalize the hyperdeterminant, and such deformations establish connections to Macdonald and Jack symmetric functions. For even $d=2m$ , the $\lambda$ -hyperdeterminant for a $2m$-way array $M$ generalizes Cayley’s object by replacing spaces of permutations in the summation with ASMs and imposing weights via simple combinatorial statistics. This formulation yields Jacobi–Trudi-type formulas for polynomials including Macdonald and Jack symmetric functions (Cai et al., 2020).

5. Entanglement Measures and Physical Applications

Cayley’s first hyperdeterminant provides polynomial measures of multipartite entanglement in quantum information. For a pure state $|\psi\rangle \in (\mathbb{C}^d)^{\otimes n}$ , interpreted as a symmetric $d \times \cdots \times d$ hypermatrix $\widehat{\psi}$ , the $2n$-partite concurrence is $2 |\mathrm{hdet}(\widehat{\psi})|$ for $d = 2$ and $n$ even. The hyperdeterminant vanishes on fully separable states and is invariant under local unitaries, satisfying all axioms of an entanglement measure, including non-increase under LOCC, both for pure and, via convex-roof extension, mixed states (Dobes et al., 22 Apr 2025, Dobes, 30 Nov 2025, Cervera-Lierta et al., 2018).

The degree-4 $2 \times 2 \times 2$ hyperdeterminant serves as the three-qubit "3-tangle," vanishing on W states and nonzero for GHZ-type states; its modulus quantifies genuine tripartite entanglement (Holweck et al., 2023). For four-qubit systems, the degree-24 Schläfli hyperdeterminant decomposes as $S^3 - 27T^2$ into invariants $S$ (degree 8) and $T$ (degree 12), refining the analysis of quadripartite entanglement and explicitly capturing the structure of eigenstates of integrable spin chains (Cervera-Lierta et al., 2018).

In supergravity and string theory, the hyperdeterminant arises in entropy formulas for extremal black holes, encoding duality and charge configuration information. The black hole entropy in the $STU$ model is given by $S_\text{BH} \sim \pi \sqrt{|\mathrm{Det}(A)|}$ , with $A$ the charge tensor (Gibbs, 2010, Holweck et al., 2023).

6. Connections to Algebraic Geometry, Number Theory, and Higher Invariants

In algebraic geometry, Cayley’s $2 \times 2 \times 2$ hyperdeterminant defines the dual variety of the Segre embedding of $(\mathbb{P}^1)^3$ into $\mathbb{P}^7$ , characterizing tensors with degenerate flattenings. The degree-24 $2 \times 2 \times 2 \times 2$ Schläfli hyperdeterminant arises as the discriminant of the quartic in projective space, and its invariants connect to the $j$ -invariant of elliptic curves (Gibbs, 2010). The presence of the hyperdeterminant in the Weierstrass equations of elliptic curves with specified torsion structures illustrates a deep interplay with arithmetic geometry.

Generalizations extend to mixed format hyperdeterminants, Spin and exceptional group invariants (notably for $E_7$ and $E_8$ ), and to fermionic and bosonic Fock spaces, producing hierarchies of discriminants that classify multipartite entanglement and are natural in representation theory (Holweck et al., 2023).

7. Rank Constraints and Combinatorial Applications

Nonvanishing Cayley hyperdeterminant imposes strong constraints on tensor rank. Specifically, for order- $d$ $n \times \cdots \times n$ tensors, the nonvanishing of the hyperdeterminant enforces that the slice rank is \emph{full}, i.e., at least $n$ , and, in positive characteristic, implies linear lower bounds on partition rank (Amanov et al., 2021). These rank lower bounds provide combinatorial tools in bounding the size of sum-free sets and evaluating generalized zero patterns in polynomials associated to high-dimensional combinatorial optimization problems.