G-Equivariant Linear Systems

• G-equivariant linear systems are constrained by Lie group symmetries, enabling geometric reduction and computational simplification.
• They facilitate classification and canonical form reduction through iterative methods and representation theory, improving control system analysis.
• Their applications span robotics, celestial mechanics, and observer design, harnessing symmetry for robust performance and efficiency.

A G-equivariant linear system is a linear system whose structure is constrained by the symmetry of a Lie group G acting on the underlying state space. Such systems arise naturally in geometry, control, representation theory, dynamical systems, and applications including robotics, celestial mechanics, and chemistry. The paper of these systems draws together deep results from differential geometry, representation theory, and systems theory, enabling symmetry-based simplification of computational and structural questions.

1. Definitions and Canonical Structure of G-Equivariant Linear Systems

Consider a Lie group G acting smoothly on a manifold M. The space of G-equivariant maps,

$$GM = \{ \xi : G \to M \mid \xi(g) = g \cdot x \text{ for some } x \in M \}$$

is in bijection with M via the canonical map $v: M \to GM$, $v(x) = \xi_x$, where $\xi_x(g) = g \cdot x$ (Aghayan, 2010). This bijection is G-equivariant for the natural induced group action: $(g, \xi)(h) = g \cdot \xi(h),$ and if G acts freely and regularly on M, then the bijection is also a diffeomorphism, so $GM$ can be endowed with a unique smooth structure such that the induced G-action is smooth.

In the context of linear systems, G-equivariant maps and operators are those that commute with the group action: for a linear operator $L: V \to V$, equivariance means $L(g \cdot v) = g \cdot L(v)$ for all $g \in G$, $v \in V$. The configuration space or solution manifold can often be expressed as $G/G_x$, the homogeneous space for some isotropy subgroup $G_x$, leveraging the canonical diffeomorphism $G/G_x \simeq M \simeq GM$ in the transitive case.

The G-equivariant structure implies that both geometric and algebraic information about orbits, stabilizers, and invariants becomes manifest in the analysis and design of linear systems, forming a foundation for symmetry reduction and computational simplification.

2. Classification, Canonical Forms, and Iterative Systems

The structure of linear systems admitting maximal symmetry (i.e., canonical G-equivariance) is tightly constrained. For a system of linear ODEs

$$\mathbf{y}^{(n)} + B_1(x)\mathbf{y}^{(n-1)} + \ldots + B_n(x)\mathbf{y} = 0,$$

the equivalence group of point transformations takes the form $x = f(z)$, $\mathbf{y} = T(z)\mathbf{w}$ (or, in “normal form,” incorporates scaling by $f'(z)^{(n-1)/2}$) (Ndogmo, 2015). The system is reducible to the canonical form $\mathbf{y}^{(n)} = 0$ precisely when, in normal form, it decomposes as repeated copies of the same scalar iterative equation.

This canonical structure is a manifestation of G-equivariance: if the system’s coefficients are scalar multiples of the identity, maximal symmetry (and maximal Lie algebra of infinitesimal symmetries) is achieved. The explicit transformation formulas guarantee that the reduction to canonical form and separation into decoupled subsystems are closed under the equivalence group, cementing the link with G-equivariant structure.

The general solution of iterative systems admits a superposition formula involving fundamental solutions of a scalar source equation, reflecting the symmetry-induced isotropy.

3. Algebraic and Operadic Characterization

G-equivariant linear systems frequently admit a classification in terms of their underlying algebraic structure. For finite G and linear systems arising from algebras over operads, any finite-dimensional simple G-equivariant algebra is of the form $A = \mathrm{Fun}_H(G, B)$, where H is the stabilizer of a minimal ideal and $B$ is a simple algebra with an H-action (Etingof, 2015).

In this setting, the system decomposes into (possibly many) copies of a simpler system with local symmetry H, then induces global G-equivariant structure through the action on functions $G \to B$: $f(g h) = \theta(h^{-1})f(g),\quad h \in H,$ with G acting by left translation. This modular construction allows for the classification of systems by identifying their building blocks up to conjugacy by G.

This perspective generalizes to the context of linear G-equivariant systems arising in infinite dimensions and in categories of modules over operads or group algebras.

4. Differential Geometry, Orbit Equivalence, and Homogeneous Spaces

The theory of G-equivariant linear systems extends into the field of geometric control and differential geometry. When the system is defined by vector fields on a smooth manifold M and G acts transitively, one may identify $M \simeq G/H$. The equivalence of control systems (as semigroup actions) on M and linear control systems on a homogeneous space $G/H$ is established when the controlled vector fields generate a full-rank, finite-dimensional Lie subalgebra and completeness/compatibility conditions are met (Hungaro et al., 2016).

The key step is constructing a diffeomorphism between the state space M and $G/H$ using the flows generated by the symmetry group. This enables reduction of arbitrary systems with symmetry to simpler representatives, facilitating analysis of controllability and stabilization through the much better-understood structure of linear invariant systems on homogeneous spaces.

The orbit equivalence preserves order and structure of control sets, rendering geometric aspects and symmetry directly accessible for both computation and classification.

5. Applications in Observer, Filter, and System Design

Modern observer and filter design for systems possessing group symmetries exploits G-equivariant structure in both the system kinematics and measurements. Consider a system evolving on a homogeneous space $\mathcal{M}$ with a transitive Lie group action; equivariance extends to inputs and outputs when the system is formulated in a symmetry-adapted fashion (Mahony et al., 2020, Goor et al., 2021).

Central concepts include:

• System Lifting: By constructing an equivariant lift $\Lambda: \mathcal{M} \times V \to \mathfrak{g}$ such that $\mathrm{d}\phi_\xi\Lambda(\xi, v) = f(\xi, v)$, one can reconstruct system trajectories on G and define observer dynamics that are invariant under group action.
• Error Dynamics: Defining global invariant errors (e.g., $e = \phi(\hat{X}^{-1}, \xi)$) ensures that error dynamics decouple from trajectory specifics and depend only on initial data and measurements.
• Equivariant Input Extension: In practical applications, when the velocity input space is not closed under the group action, it can be extended to guarantee global equivariance.
• Filter Linearization: By linearizing error dynamics globally at a fixed origin, one obtains system and observer matrices that are independent of the current estimated state, dramatically reducing computational burden and increasing stability domains.

These geometric and algebraic methods provide a systematic pathway to design robust, globally consistent state estimators, applicable to a wide array of mechanical, aerospace, and robotic systems.

6. Computational Reduction and Block-Diagonalization

Symmetry-induced block-diagonalization is a critical computational tool. When an operator M on a vector space V commutes with the G-action, representation theory ensures V decomposes into isotypic components, and M can be conjugated into block-diagonal form via a symmetry-adapted basis (Silva et al., 23 Sep 2025). Projection operators constructed from the group's irreducible representation characters,

$$P_{kl}^{(j)} = \frac{n_j}{|G|}\sum_{g \in G} d^{(j)}_{lk}(g^{-1}) \phi(g),$$

yield explicit algorithms for this transformation.

The result is a partition of the original operator into blocks associated with each irreducible G-subrepresentation, each of which is substantially smaller than the original operator. This has practical ramifications—eigenvalue problems, matrix exponentials, and symbolic manipulations can all proceed blockwise, vastly increasing computational efficiency and enabling symbolic analysis in intractable cases (e.g., symbolic celestial mechanics, quantum systems).

7. Observability and Symmetry-Induced Invariants

G-equivariant structure facilitates separation of observability and computational components. By leveraging the unique diffeomorphism $v: M \to GM$, system orbits correspond bijectively to equivalence classes of the action (Aghayan, 2010). The induced manifold structure on GM and the resulting smooth structure on $M/G$ provide a rigorous setting to analyze which states are distinguishable by equivariant measurements and which are indistinguishable due to group symmetry.

In control and estimation, this framework leads to significant simplification of observability analysis: the focus can be shifted to the quotient space, where observables are invariants under G, reducing nonlinear problems to tractable linear problems on the quotient or cross-section.

8. Broader Implications and Theoretical Significance

G-equivariant linear systems encode the fundamental principle that symmetry dictates structure. Their classification and reduction theory encompass canonical forms for ODEs, modular decomposition via induced algebra construction, orbit-equivalence reduction to homogeneous spaces, and systematic computational simplification. The interplay between group actions, system theory, and algebraic geometry underlies developments in observer design, symmetry reduction in large-scale computational physics, and analytical results in invariant theory and algebraic geometry.

Uniqueness of equivariant structures guarantees robust and predictable behavior of both theoretical constructions and applied algorithms, ensuring that any symmetry-exploiting methodology developed in this context retains fidelity to the underlying system's invariants and symmetries.