SL-Invariant Homogeneous Polynomials

• SLIPs are homogeneous polynomials invariant under special linear group actions, existing only when each n_i divides the polynomial degree.
• Degree bounds and explicit generators, such as determinants and hyperdeterminants, play a key role in characterizing SLIPs in tensor and matrix settings.
• Applications in algebraic complexity and quantum information utilize SLIPs to analyze orbit closures, design invariant-based algorithms, and distinguish multipartite entanglement classes.

A homogeneous SL-invariant polynomial (SLIP) is a polynomial function on a representation space that is homogeneous of given degree and remains invariant under the action of a special linear group (or a product thereof). SLIPs are central to classical invariant theory, the theory of polynomial invariants of tensors and matrices, the study of orbit closures and their null-cones, and applications in algebraic complexity and quantum information theory.

1. Definitions and General Structure

Let $$G = \text{SL}(n_1) \times \cdots \times \text{SL}(n_d)$$ act linearly on the tensor space $$V = \mathbb{C}^{n_1} \otimes \cdots \otimes \mathbb{C}^{n_d}$$. The associated coordinate ring $\mathbb{C}[V]$ is graded by total polynomial degree. An element $f \in \mathbb{C}[V]$ is called a homogeneous SL-invariant polynomial (SLIP) of degree $m$ if it is homogeneous of degree $m$ and $g \cdot f = f$ for all $g \in G$. The subspace of such invariants of degree $m$ is denoted $\operatorname{Inv}(n_1,\dots,n_d)_m$ (Amanov et al., 2022).

A fundamental property is that nontrivial invariants can exist only if $n_i \mid m$ for all $i$ (a consequence of the determinant condition on $\text{SL}(n_i)$ actions). The graded subring of SLIPs is finitely generated due to reductivity of $G$.

2. Degree Bounds and Fundamental Invariants

For tuples of $d$-mode tensors, the minimal degree $m_0 = \delta(n_1,\ldots,n_d)$ for which a nonzero invariant exists is called the degree of the fundamental invariant. In the classical case $d = 2$ and $n_1 = n_2 = n$, $\delta_2(n) = n$ and the single fundamental invariant is the determinant.

For general $d$ and $n_1 = ... = n_d = n$, the situation is more complex:

• For even $d$, $\delta_d(n) = n$ (Cayley's hyperdeterminant).
• For odd $d \geq 3$, the minimal degree satisfies $n\lceil n^{1/(d-1)}\rceil < \delta_d(n) < n^2$, and the lower bound is sharp for infinitely many $n$ (Amanov et al., 2022).

Explicit constructions use polytabloid-type formulas—sum over particular combinatorial structures indexing the terms, such as Latin hypercubes, with each term weighted by suitable signs. For $d=3$, a known example is Cayley’s 2×2×2 “hyperdeterminant”. The uniqueness and existence of such minimal degree invariants is often linked to deep combinatorial conjectures (see Section 5).

3. Structure Theorems: Generation and Null Cone

The ring of SLIPs in many contexts has its structure determined by degree bounds for generators and the null-cone. Consider the case of $m$-tuples of $n \times n$ matrices under the "left-right" action of $\text{SL}_n \times \text{SL}_n$. The coordinate ring $$R(n,m) = \mathbb{K}[V]^{\text{SL}_n \times \text{SL}_n}$$ is graded by degree $d$, and $R(n,m)_d = 0$ unless $n \mid d$ (Derksen et al., 2015).

Key results:

• The null-cone is defined set-theoretically by SLIPs of degree at most $n^2 - n$.
• The entire ring $R(n,m)$ is generated by invariants of degree $\leq n^6$ if $\operatorname{char}(K) = 0$.
• For large $m$, invariants of degree at least $n \lfloor \sqrt{n+1} \rfloor$ are necessary to define the null-cone.
• These results generalize to semi-invariants of quivers as well as to spaces of rectangular matrices (Derksen et al., 2015).

The proofs leverage "tensor blow-up" constructions and properties such as the regularity lemma and concavity of matrix ranks with respect to blow-up parameters.

4. Representation-Theoretic Constructions and Hilbert Series

The computation and enumeration of SLIPs are governed by Schur–Weyl duality and representational methods. In the case of multipartite quantum systems, for the space $$\mathcal{H}_n = \mathbb{C}^{d_1} \otimes \dots \otimes \mathbb{C}^{d_n}$$, all degree-$k$ SLIPs are the $G$-fixed points in the $k$-fold tensor power: $((\mathcal{H}_n)^{\otimes k})^G$ (Gour et al., 2013). This space can be constructed explicitly via central idempotent projectors coming from the symmetric group $S_k$.

The dimension of the space of degree-$k$ invariants corresponds to a sum over $S_k$-characters: $$\dim I_k = \frac{1}{k!} \sum_{\sigma \in S_k} [\chi_\lambda(\sigma)]^n,$$ where, for qubits $(d_j = 2)$, $k$ even, $\lambda = (k/2, k/2)$. This machinery gives explicit enumeration and formulas for Hilbert series.

For $\text{SL}_2$-invariants, Hilbert series computations employ the Molien–Weyl integral, yielding rational expressions and closed formulas for Laurent expansion coefficients of the series at $t=1$, encapsulating deep structural invariants and dimension counts (Pinto et al., 2017).

5. Explicit Generators and Characteristic Examples

The explicit description of generators is essential in both classical and quantum information contexts. For four qubits, the set of $\text{SL}(2, \mathbb{C})^{\otimes 4}$-invariant, permutation-symmetric, homogeneous polynomials is freely generated by four algebraically independent invariants of degrees 2, 6, 8, and 12, constructed via coordinates in a suitably chosen critical-state basis. The unique (up to scalar) invariant vanishing precisely on non-generic orbits is the hyperdeterminant of degree 24 (Gour et al., 2012). For binary forms, explicit generators and minimal syzygies are constructed via transvectants, discriminants, and resultants (Brouwer et al., 2011).

Representative examples from geometric complexity theory include the determinant, the permanent, the simple product, power sums, the unit tensor, and the matrix multiplication tensor, for which fundamental invariants often coincide with minimal SLIPs determined by tableau or Young symmetrizer techniques. For the determinant and the permanent, the existence and properties of the minimal degree fundamental invariant are linked to combinatorial sign sums over admissible tableaux analogous to the Alon–Tarsi conjecture (Bürgisser et al., 2015).

6. Combinatorics, Kronecker Coefficients, and Alon–Tarsi Type Conjectures

The existence and nonvanishing of minimal-degree SLIPs are frequently governed by combinatorial identities and conjectures. For tensors with $d$ modes, the value of the minimal degree and even the existence/nonvanishing of the fundamental SLIP may depend on Alon–Tarsi type counts over Latin hypercubes (multi-dimensional analogues of Latin squares) (Amanov et al., 2022). Positivity of generalized or rectangular Kronecker coefficients is shown to be equivalent to the existence of such invariants in specific degrees. For $d$ odd and $k$ even, the nonzero value of the $d$-dimensional Alon–Tarsi sum $AT_d(k)$ guarantees positivity of the Kronecker coefficient $g_d(n, k)$ for all $n < k^{d-1}$, determining the degree of the fundamental invariant in this range.

These combinatorial conditions also manifest in geometric complexity theory, relating the existence of explicit SLIPs for orbit closures to open enumerative and sign-balance problems (Bürgisser et al., 2015).

7. Applications in Complexity and Quantum Theory

Sharp degree bounds and explicit knowledge of SLIPs have direct algorithmic and physical applications. In algebraic complexity theory, degree bounds for matrix semi-invariants underpin deterministic polynomial-time algorithms for noncommutative rational identity testing, and support the construction of division-free noncommutative arithmetic circuits. The degree lower bounds further imply exponential lower bounds on the size of certain circuits computing noncommutative determinants (Derksen et al., 2015).

In quantum information, SLIPs provide the complete system of invariants distinguishing SLOCC classes for pure multipartite states. Ratios of homogeneous SLIPs in the amplitudes of a quantum state remain constant on SLOCC orbits, and the explicit generators quantify genuine multipartite entanglement. For example, the algebraic structure of SLIPs for four qubits evidences the infinite richness of SLOCC classes compared to two- or three-qubit systems (Gour et al., 2012, Gour et al., 2013).