SO(2)-Equivariant Operations

SO(2)-equivariant operations are mathematical constructions, transformations, or operators that commute with the continuous group of planar rotations denoted SO(2). Such operations play a fundamental role across mathematics, physics, and machine learning, providing the means to encode rotational symmetry into models, operators, or physical theories. An operation $L$ acting on a suitable space is SO(2)-equivariant if $L[R(\theta) x] = R(\theta) L[x]$ for all $\theta$, where $R(\theta)$ is the action of SO(2) (a rotation by angle $\theta$) on the relevant space. SO(2)-equivariance appears in contexts ranging from representation theory and quantum physics to equivariant neural networks, cohomology theories, and computational chemistry.

1. Mathematical Framework and Principles

SO(2) is the group of rotations in two dimensions and serves as a prototype for continuous symmetries. The defining property for an SO(2)-equivariant operation is that it commutes with these rotations. Mathematically, if $V$ and $W$ are vector spaces equipped with representation of SO(2), then a map $T: V \rightarrow W$ is SO(2)-equivariant if $T(\rho_v(g) v) = \rho_w(g) T(v)$ for all $g \in SO(2)$. In the functional setting, this often reduces to requiring that $T$ intertwines the angular Fourier (or circular harmonic) modes, such as: $$T\big( e^{ik\theta} \big) = \lambda_k e^{ik\theta} \, , \qquad k \in \mathbb{Z}.$$ For tensorial fields, equivariant operations must also respect the transformation ranks, meaning their action preserves or raises/lowers the angular momentum index in a manner dictated by angular convolution rules.

A key implementation pattern is to represent quantities in a basis diagonalizing the SO(2) action—e.g., the Fourier basis on $[0, 2\pi]$, or for functions on $\mathbb{R}^2$, using polar coordinates and complex exponentials $e^{ik\theta}$. SO(2)-equivariant operators then act algebraically in Fourier space, generally as block-diagonal or banded operators.

2. SO(2)-Equivariant Operations in Physics and Representation Theory

In mathematical physics, SO(2)-equivariant operations encode rotational symmetry in both quantum and classical settings.

• In the theory of angular momentum (quantum mechanics), SO(2) equivariance ensures conservation of the angular momentum component. In Hilbert space, basis functions $$\psi_m(\theta) = \frac{1}{\sqrt{2\pi}} e^{-im\theta}$$ are eigenfunctions of the angular momentum operator $J = -i \frac{d}{d\theta}$, and all physically meaningful operators—such as the Hamiltonian of a rotationally symmetric system—must be SO(2)-equivariant; that is, diagonal or banded in this representation (Celeghini et al., 2017 ).
• Rigged Hilbert spaces ($\Phi \subset H \subset \Phi^\times$) provide a setting for accommodating both the discrete (Fourier) basis and generalized eigenstates, with group actions continuously extended throughout. SO(2)-equivariant operators act naturally on this triplet, ensuring mathematical validity for both bounded and generalized operations in signal analysis and quantum physics.

In geometric representation theory and harmonic analysis, SO(2)-equivariant operations appear in the classification and integration over orbits, such as the paper of orbital measures and SO(2)-bi-invariant convolutions. On the symmetric space $SU(2)/SO(2)$, orbital measures define SO(2)-bi-equivariant integral operators whose properties (such as smoothing or regularity under convolution) are governed by the interplay between SO(2) symmetry and the harmonic decomposition of functions (Anchouche et al., 2014 ).

3. Algebraic and Cohomological Models of SO(2)-Equivariant Operations

In abstract homotopy theory and algebraic topology, SO(2)-equivariant operations are rigorously modeled with algebraic machinery:

• The homotopy category of rational SO(2)-equivariant spectra admits a Quillen equivalence to a differential graded abelian category $d\mathcal{A}(SO(2))$ (Barnes et al., 2015 ). Objects therein are algebraic structures (triples of graded modules and structure maps) encoding the possible SO(2)-fixed and isotropy data of a spectrum. Equivariant operations, including maps, module structures, or (naive) commutative products, are then homological algebra constructs—computable via Hom and Ext groups—thus reducing topological equivariance to algebraic calculation.
• In rational equivariant elliptic cohomology, commutative algebras in the algebraic model correspond to $E_\infty$-algebras in the spectrum category, with SO(2)-equivariant operations encoded as module maps and sheaf-theoretic transformations on associated algebraic curves (Barnes et al., 2018 ). The derived category of modules over such spectra equates to the derived category of sheaves over the elliptic curve, reflecting a deep paradigm linking geometric and algebraic symmetry via SO(2)-equivariance.
• In algebraic geometry, the computation of multiplicative operations on T-equivariant oriented cohomology (where T may be SO(2)) is framed through formal group laws and Schubert calculus techniques, realizing equivariant cohomology operations as explicit algebraic maps that commute with the torus action, up to known twists given by inverse Todd classes (Zainoulline, 2020 ).

4. SO(2)-Equivariant Operations in Machine Learning and Signal Processing

SO(2)-equivariance is foundational in the construction of neural network architectures and learning models that respect planar rotation symmetry:

• In convolutional neural networks, SO(2)-equivariant operations can be realized by representing features and filters in a radial-harmonic (e.g., Bessel or Fourier-Bessel) basis. In Bessel-convolutional neural networks (B-CNNs), for example, both images and filters are decomposed into Fourier-Bessel modes; rotations act as explicit phase shifts on those coefficients, allowing for exactly SO(2)-equivariant transformations throughout the network (Delchevalerie et al., 2023 ). The key operation

$$a(x, y) = \sum_\nu \left| \sum_j \kappa_{\nu, j}^* \varphi_{\nu, j}^{(x, y)} \right|^2$$

is rotation-invariant at each patch and therefore globally SO(2)-equivariant.

• For vector and tensor field networks, such as those used in physics-informed learning or geometric deep learning, SO(2)-equivariant operators are implemented as generalized convolutions with radially symmetric kernels,

$$L(\mathbf{u}) = \mathbf{u} * \left( R(|\mathbf{r}|) \mathbf{Y}_l(\hat{\mathbf{r}}) \right)$$

ensuring that the operation commutes with arbitrary planar rotations (Shen et al., 2021 ).

• In SO(2)-equivariant neural architectures, nonlinearities, normalization, tensor products, and gate structures are designed to act block-wise on angular Fourier modes or irreducible SO(2) representations. This enables the construction of expressive, symmetry-preserving layers required in equivariant networks for tasks ranging from 2D/3D vision to quantum chemistry.

The challenge of nonlinear activations, which may introduce higher angular frequencies, is tackled by using FFT-based algorithms that apply nonlinear functions pointwise in the angular domain, then project back to the limited frequency band. This enables continuous SO(2) equivariance even under standard nonlinearities, a property validated in both theory and practice (Franzen et al., 2021 ).

5. Reduction of SO(3)-Equivariant Operations to SO(2) via Local Frames

Many physical, chemical, and geometric problems are inherently SO(3)-equivariant but can be efficiently implemented using SO(2)-equivariant operations by aligning local frames to geometry.

• In recent geometric deep learning models for 3D data (e.g., molecule modeling, point cloud analysis, and quantum chemistry), global SO(3) equivariance is achieved by expressing features in SO(2) frames aligned to object pairs (e.g., atomic pairs) or edge vectors (Yu et al., 11 Jun 2025 , Passaro et al., 2023 ). Operations such as message passing or feature update then occur within these local SO(2) frames using SO(2)-equivariant layers—linear, nonlinear, normalization, or tensor product—encoded in the angular momentum basis.
• Tensor products and contractions that would be implemented using high-cost SO(3) Clebsch–Gordan algebra can be replaced with SO(2) tensor products inside these frames, greatly reducing computational complexity (from $O(L^6)$ to $O(L^3)$ or even lower).
• Frame averaging (recombining local-frame results into the global coordinate system) preserves overall SO(3) equivariance, as shown in explicit formulas:

$h \cdot \Phi(h^{-1} \cdot \hat{r})$

where $h$ defines the alignment and $\Phi$ is an SO(2)-equivariant map.

This reduction enables state-of-the-art scalability and performance in quantum Hamiltonian matrix prediction, 3D GNNs, and other equivariant learning settings by leveraging the tractability and power of SO(2) representations (Yu et al., 11 Jun 2025 , Passaro et al., 2023 ).

6. Classification and Construction of SO(2)-Equivariant Operations

A general algebraic approach to SO(2)-equivariant maps considers the space of equivariant maps from a homogeneous space $G/H$ to a module $V$: $\mathrm{Hom}_{G}(G/H, V) \cong V^{H}$ where $V^{H}$ is the subspace of $H$-invariant vectors. For $SO(2)$, this means that all equivariant operations (e.g., convolution kernels, pointwise nonlinearities, biases) can be characterized by their invariance under the action of SO(2). When the module is a space of matrices, this leads to block-diagonal or structured algebraic forms—facilitating efficient and computable neural layers (Knibbeler, 2023 ).

Automorphic algebra techniques further classify the algebraic structure of SO(2)-equivariant operators, showing that their parameter spaces decompose into $\mathfrak{gl}(r, \mathbb{R})$ and $\mathfrak{gl}(c, \mathbb{C})$ blocks according to the symmetry types present in the representation. This systematic approach guides both the theoretical and algorithmic construction of all possible SO(2)-equivariant operations in practice.

7. Applications and Implications

SO(2)-equivariant operations underpin a diverse spectrum of applications:

• Physics: Modeling rotationally symmetric systems, enforcing angular momentum conservation, and recognizing the correct universality classes in field theory (e.g., Gross-Neveu-Yukawa models with tensor order parameters, where the symmetry controls critical exponents and gap ratios) (Han et al., 25 Nov 2024 ).
• Mathematics: Facilitating explicit harmonic analysis, characterizing spaces of orbital measures, and understanding representation-theoretic invariants.
• Topological and algebraic computations: Providing tractable algorithms for equivariant cohomology, sheaf computation, and spectrum/module categories.
• Machine Learning and Computer Vision: Enabling exact or approximate rotation-equivariant neural architectures for images, fields, graphs, and molecules; improving generalization without data augmentation.

Empirical studies in quantum chemistry, geometric deep learning, and image classification show not only improved accuracy and generalization but also substantial computational savings when SO(2)-equivariant constructions are explicitly leveraged, especially via local-frame or spectral decompositions.

Summary Table: Key Instances of SO(2)-Equivariant Operations

SO(2)-equivariant operations, defined by commutation with the planar rotation group action, form a structural foundation for symmetry-aware computation and analysis in mathematics, theoretical and applied physics, and modern data science. Their design, implementation, and classification enable efficient, robust, and interpretable models that respect the rotational invariances present in both natural and engineered systems.