Rotation Invariant Trilinear Forms

• Rotation invariant trilinear forms are mappings that remain unchanged under rotations, serving as key invariants in representation theory and harmonic analysis.
• They are constructed using advanced tools such as group representations, analytic continuation, and kernel decomposition, ensuring precise handling of singular integrals.
• These forms have diverse applications, impacting quantum information, algebraic geometry, and particle physics by constraining invariant functionals and geometric structures.

A rotation invariant trilinear form is a trilinear mapping or distribution that remains unchanged under rotations, often arising in settings with high symmetry, such as representation theory, harmonic analysis, quantum information, algebraic geometry, and particle physics. These forms capture the essential algebraic, analytic, or combinatorial constraints imposed by rotation (usually SO(n)), or larger symmetry groups, and are frequently involved in the classification of invariant functionals, geometric invariants, or singular integrals.

1. Definitions, Symmetry, and General Frameworks

A trilinear form $\mathcal{T}$ on a vector space $V$ (or collection of spaces $V_i$) is said to be rotation invariant if for all rotations $R$ (typically elements of SO(n)), and all $f_1, f_2, f_3 \in V$, one has

$$\mathcal{T}(R f_1, R f_2, R f_3) = \mathcal{T}(f_1, f_2, f_3).$$

In practice, such forms may be invariant under larger groups (full conformal group, quantum groups, or projective symmetries), with rotation invariance being a necessary sub-property. In harmonic analysis or representation theory, these forms naturally appear as invariants under the diagonal action of symmetry groups, e.g., $G$ acting on $V_1 \otimes V_2 \otimes V_3$ via $(g f_1, g f_2, g f_3)$.

The construction of rotation invariant trilinear forms frequently exploits:

• Group representations, e.g., spherical principal series in conformal geometry, minuscule or highest weight representations in algebraic combinatorics, or induced representations in automorphic forms.
• Analytic continuation or meromorphic extension (to handle singular cases or “resonant” parameters).
• Explicit integration over geometric domains (spheres, flag varieties, etc.) with invariant kernels.
• Kernel decompositions or reduction of high-dimensional problems via “method of rotations” (Calderón–Zygmund theory).

2. Conformally Invariant and Rotation Invariant Trilinear Forms on the Sphere

A paradigmatic instance occurs in the paper of conformally invariant trilinear forms on the sphere $S^{n-1}$, as in the principal series of $SO_0(1,n)$ (Clerc et al., 2010, Clerc, 2011, Clerc, 2014, Clerc, 2015). For each complex $\lambda$, consider the representation on smooth functions

$$[\pi_\lambda(g) f](x) = \kappa(g^{-1}, x)^{\lambda+\rho} f(g^{-1} x), \qquad \rho = (n-1)/2,$$

where $\kappa$ is the conformal factor, with the property that the action of rotations SO(n) is independent of $\lambda$ (i.e., rotation invariance is always present).

For triples $(\lambda_1, \lambda_2, \lambda_3)$, construct a kernel $K_\alpha(x_1,x_2,x_3)$, with $\alpha$-parameters determined by explicit linear combinations of the $\lambda_j$. The trilinear form is then

$$\mathcal{T}(f_1, f_2, f_3) = \int_{S \times S \times S} K_\alpha(x_1, x_2, x_3) f_1(x_1) f_2(x_2) f_3(x_3) d\sigma(x_1)\,d\sigma(x_2)\,d\sigma(x_3).$$

This form is invariant under the diagonal action of the conformal group and, in particular, is SO(n)-invariant. Its essential properties are:

• Meromorphic extension: The form extends analytically in the parameters to all of $\mathbb{C}^3$, with simple poles on explicit hyperplanes.
• Multiplicity one theorem: For generic parameter choices (away from the pole sets), the invariant space is one-dimensional; i.e., conformal invariance fully constrains the trilinear form up to scaling, and rotation invariance introduces no extra degrees of freedom (Clerc et al., 2010, Clerc, 2014).
• Singular locus and residues: At the poles (e.g., when a spectral parameter hits $-\rho - 2k$), residues produce new, “singular” invariants, often constructed via covariant differential operators (Clerc, 2011, Clerc, 2015).

3. Higher Multiplicity Cases and Orbit Decompositions

In higher rank or singular limit settings, the dimension of the space of invariant trilinear forms may jump. For $SL(3,\mathbb{R})$, the trilinear forms associated with the triple product of the flag variety $(P\backslash G)^3$ are essentially governed by the orbit structure under the diagonal action (Deitmar, 2018). There exist both discrete and continuous families of $G$-orbits, leading to an infinite-dimensional space of invariant trilinear forms when no open $G$-orbit exists. The rotation invariance per se is always present, but for such higher rank groups, the full orbit analysis is essential for describing the space of invariants.

4. Harmonic Analysis and Singular Integral Trilinear Forms

Rotation invariant trilinear forms arise centrally in harmonic analysis, particularly as multilinear singular integrals with maximal symmetry. A prominent example is (Gressman et al., 2015), where the trilinear form

$$A(f,g,h) = \mathrm{p.v.} \int_{\mathbb{R}^2} f(x)g(y)h(z)\delta(x+y+z)\,\det\begin{pmatrix}1 & x \ 1 & y \ 1 & z\end{pmatrix}^{-1} dx\,dy\,dz$$

is shown to be invariant under the group of all invertible $2\times 2$ real matrices (GL$_2(\mathbb{R})$), which contains SO(2) as a subgroup. This invariance guarantees that $A$ is structurally “rotation invariant”. The full symmetry of $A$ enables an exact determination of the exponents $2 < p, q, r < \infty$ with $1/p + 1/q + 1/r = 1$ for which $A$ is bounded from $L^p \times L^q \times L^r$ to $\mathbb{C}$. Reducing such forms to superpositions of bilinear Hilbert transforms and their relation to the Calderón commutator depends crucially on the rotation invariance of the underlying kernel.

The broader theory of trilinear singular Brascamp–Lieb forms, as classified in (Becker et al., 31 Oct 2024), not only establishes a comprehensive algebraic module-theoretic organization but also develops a “method of rotations”—decomposing homogenous Calderón–Zygmund kernels in $\mathbb{R}^d$ into lower-dimensional (rotation-invariant) slices on hyperplanes. Given a kernel $K$ with Fourier transform $\widehat{K}(\xi) = \Omega(\xi/|\xi|) + C_0$, this approach allows one to express $K$ as a superposition

$$K(x) = \int_{S^{d-1}} K_\theta(x)\, d\sigma(\theta),$$

with $K_\theta$ being lower-dimensional Calderón–Zygmund kernels on hyperplanes orthogonal to $\theta$. This “method of rotations” leverages the fundamental rotation invariance to reduce high-dimensional analysis to tractable lower-dimensional forms, providing a powerful mechanism for proving $L^p$ bounds in the multilinear singular integral context.

5. Quantum Information and Operator Space Theory

In quantum information and operator space theory, rotation invariant trilinear forms play a role in the analysis of Bell inequalities and on the geometry of tensor norms (Pisier, 2012). For example, consider trilinear forms on $\ell_1^n \otimes \ell_1^n \otimes \ell_1^n$ with coefficients drawn from the second Gaussian Wiener chaos. The norms studied—the injective norm $\|\cdot\|_{\vee}$ and the minimal norm $\|\cdot\|_{\min}$—are both naturally compatible with the unitary (rotation) invariance of the Gaussian measure. The Briët–Vidick method establishes that

$$\|I_n: E^n_{\vee} \rightarrow E^n_{\min}\| \geq c n^{1/4} (\log n)^{-3/2},$$

exploiting the rotation invariance both in the probabilistic construction of random tensors and in the structure of operator spaces.

6. Applications in Algebraic Geometry, Schubert Calculus, and Physics

Symmetric trilinear forms with rotation invariance also arise as intersection forms in algebraic geometry. In Calabi–Yau threefolds, the trilinear cup product on $H^2(X, \mathbb{Z})$ is symmetric and O($b_2(\mathbb{R})$)-invariant; if the associated cubic form factors as a product of a linear and quadratic form, the residual quadratic structure is preserved under orthogonal transformations (rotations) and constrains the possible group actions (Kanazawa et al., 2012).

In the context of Schubert calculus, the local combinatorial “puzzle” rules on flag manifolds can be reinterpreted via vector configurations from minuscule representations, where each puzzle piece encodes a trilinear invariant—these are precisely the invariant trilinear forms that sew together to produce global Schubert structure constants. The triality symmetry in $D_4$ (2-step case), for example, manifests as a $120^\circ$ rotation among minuscule representations, and the “bootstrap” property of R-matrices in quantum integrable models degenerates to the invariant trilinear form, see (Knutson et al., 2017).

In particle physics, rotation-invariant trilinear forms appear as trilinear Higgs self-couplings in extended Higgs sectors with discrete symmetry (e.g., $S(3)$-invariant models). These couplings acquire their structure via rotation matrices that diagonalize scalar mass matrices, and their analytical forms depend explicitly on rotation angles in scalar field space (Barradas-Guevara et al., 2014).

7. Summary Table: Rotational Invariant Trilinear Forms Across Domains

Domain	Paradigm/Example	Invariance Group
Conformal geometry	$\int K_\alpha(x_1,x_2,x_3) f_1 f_2 f_3$ on $S^{n-1}$	SO(n), SO$_0(1,n)$
Harmonic analysis	Singular integrals with determinantal kernel	GL$_2(\mathbb{R})$, SO(2)
Operator space theory	Norms of Gaussian chaos trilinear tensors	U(n), O(n)
Algebraic geometry	Cup product $\mu(x,y,z)$ on $H^2$	O(q) (orthogonal group)
Schubert calculus	Tensor contractions of minuscule representations in puzzles	Weyl, rotation/triality
Physics (Higgs sector)	Trilinear couplings after basis rotation	Physical SU(2), S(3)

8. Concluding Remarks

Rotation invariant trilinear forms provide an organizing principle in a range of mathematical settings where symmetry, representation theory, and analysis intersect. Whether as unique (multiplicity one) invariants determined by group symmetry, as structural building blocks in quantum information and combinatorics, or as operators encoding geometric or physical interactions, the constraints imposed by rotation invariance remain fundamental. The technical methods—analytic continuation, orbit decomposition, kernel decomposition via the method of rotations, and algebraic module theory—reflect the depth and sophistication required to analyze and classify these forms. Such analysis continues to drive progress in harmonic analysis, representation theory, mathematical physics, and algebraic geometry.