invariant theory

This work investigates the algebraic–combinatorial mechanisms underlying graph isomorphism discrimination and introduces the framework of “separating modules,” a polynomial vector space grounded in the representation theory of symmetric groups. By employing complexity measures such as support size, symmetric circuit size, and multiplicity, it establishes equivalences with subgraph counting (support size \(k\) corresponds to order \(O(k)\)) and the Weisfeiler–Leman algorithm (circuit size \(n^{\Theta(k)}\) corresponds to \(\Theta(k)\)-WL). The central contribution provides the first intrinsic characterization of multiplicity separation: two graphs are distinguishable if and only if their automorphism groups have distinct cycle indices. Furthermore, the paper demonstrates that the multiplicity barrier is strictly stronger than the occurrence barrier and connects invariant polynomials to the graph reconstruction conjecture and finite-type invariants.

This paper addresses the problem of determining whether a given algebraic variety (V) arises as the Zariski closure of an orbit of a point under the action of an (s)-generated commutative matrix group. To resolve this, the authors first formulate and solve the “decidability” problem for such orbit closures, establishing a unified framework integrating commutative algebra, structural theory of matrix groups, lattice theory, and algebraic-geometric analysis of orbit closures. They devise a decision algorithm that, given (V) and (s), determines in PSPACE whether (V) equals the orbit closure of some point under an (s)-generated commutative linear algebraic group. Moreover, they prove that this problem is PSPACE-complete—establishing a tight complexity characterization. The main contribution is the first proof of computability for this geometric decision problem, together with an optimal complexity bound, thereby filling a fundamental theoretical gap in the structural decidability of orbit closures under algebraic group actions.

This work addresses the problem of modeling invariant functions over symmetric matrices (under conjugation by permutations) and point clouds (under rotations, reflections, and point permutations). Methodologically, inspired by Galois theory, we construct the first lightweight universal approximator framework that yields separating invariant features of dimensionality only $O(n^2)$ for symmetric matrices and further optimizes to $O(n)$ for point clouds—breaking the bottleneck of traditional high-dimensional invariant representations. Our theoretical foundation integrates invariant algebra and generating sets of rational function fields, coupled with DeepSets architecture and orbit-separation analysis under group actions. Experiments on molecular property regression and point cloud distance prediction empirically validate almost-everywhere orbit separation, enabling universal approximation of weighted graph functions. The proposed framework significantly enhances both expressive power and computational efficiency of invariant representations.

This paper addresses the unified modeling of symmetries in machine learning. It proposes a framework grounded in differential geometry and convex optimization to (1) enforce known symmetries, (2) automatically discover unknown symmetries in models or data, and (3) actively induce symmetry breaking via user-specified candidate groups. The core contribution is the first formulation of symmetry imposition and discovery as dual linear-algebraic tasks, leveraging the Lie derivative to characterize fiberwise linear Lie group actions on vector bundles, and employing nuclear-norm relaxation to construct convex regularization terms. The method is broadly applicable to neural networks, dynamical system discovery, basis-function regression, and neural operators. Empirically, it significantly improves generalization performance and parameter efficiency—particularly in low-data regimes—while preserving geometric structure and interpretability.

This work investigates the computational complexity of the Tensor Orbit-Closure Intersection (TOCI) problem under group actions, which underlies fundamental questions in group-action equivalence testing and separability of invariant polynomials. Method: We introduce the algebraic complexity class TOCI and develop a novel algebraic reduction framework grounded in geometric invariant theory and representation theory. Contributions: We establish that tensor network equivalence is TOCI-complete—the first completeness result for TOCI. We further construct a polynomial-time many-one reduction from Graph Isomorphism (GI) to TOCI, proving GI ⊆ TOCI and thereby providing the first rigorous lower bound for TOCI. Moreover, our framework unifies and explains the complexity of longstanding open problems—including noncommutative Polynomial Identity Testing (PIT) and quantum state classification—by reducing them to TOCI. This work constitutes the first systematic complexity-theoretic treatment of orbit-closure intersection problems.

This work develops a novel algorithmic information theory within the framework of symmetric groups to characterize string complexity induced by symmetries. By introducing symmetry groups generated by computable bijections, the authors define a “symmetric prior” and, under the fix-retractable condition, prove it constitutes a universal lower-semicomputable semimeasure, thereby establishing a geometric coding theorem. The central innovation lies in the first unified integration of algorithmic information theory with group theory, proposing a new paradigm for complexity measures grounded in symmetry. Furthermore, the study reveals a structural correspondence between subgroups and sets of binary strings via a Galois connection. This theoretical foundation advances computational algorithmic statistics (CAS) and opens new avenues for analyzing structured data through algebraic and informational lenses.

This study investigates the invariance properties of distributional divergence measures within group-symmetric statistical models. By endowing both the sample and parameter spaces with a group action and assuming that density functions transform according to a multiplier representation, the authors integrate tools from group representation theory, transformation models, $f$-divergence analysis, and Fisher–Rao information geometry. They establish that all $f$-divergences and the Fisher–Rao distance are invariant under the induced group action. The key contribution lies in showing that such invariant divergences reduce to functions depending solely on the maximal invariants of the parameter pair. This framework is successfully extended to multivariate location-scale families, where the invariant geometric structure of the parameter space is characterized via double coset decompositions.

Existing equivariant neural fields struggle to handle inconsistent group actions on heterogeneous product spaces. This work proposes an isotropy subgroup reduction framework that establishes an orbit equivalence $(X \times M)/G \cong X/H$, thereby transforming the learning of $G$-invariant functions over the product space into learning $H$-invariant functions solely on $X$, where $H$ is the isotropy subgroup. By circumventing the stringent structural constraints on group actions imposed by prior methods, this approach significantly enhances modeling flexibility while preserving expressive capacity. It achieves, for the first time, a unified equivariant modeling framework applicable to arbitrary group actions and homogeneous configuration spaces.

This study addresses the long-standing lack of explicit formulas and structural understanding of Chern classes expressed as symmetric polynomials across various bases of symmetric functions. By integrating multiple artificial intelligence systems with human mathematical insight, we establish a collaborative workflow that closes the research loop from experimental exploration and conjecture generation to symbolic proof. We demonstrate for the first time the feasibility of AI-augmented pure mathematical discovery, providing explicit expressions for the Chern and K-theoretic classes of $\mathrm{Sym}^d(\mathbb{C}^n)$. Furthermore, we prove refined positivity and a novel form of log-concavity for their Schur coefficients when expanded in the binomial basis, uncovering deep combinatorial structures in the rank-two case.

Existing 2D continuous representations struggle to simultaneously preserve continuity and satisfy arbitrary plane group symmetries, particularly because non-reflection operations often disrupt continuity. This work proposes the first general-purpose symmetrization framework that rigorously enforces full plane group symmetry—including non-reflection operations—while maintaining continuity in 2D continuous representations. By integrating group-theoretic modeling with approximation theory for continuous functions, the method transforms any 2D continuous representation into one that strictly adheres to prescribed symmetries without compromising smoothness. The approach is validated across four diverse applications: pattern design, kirigami art, stylized topology, and material design, demonstrating high-fidelity, controllable generation of symmetric patterns. This study thus achieves, for the first time, full compatibility between general plane group symmetries and continuous 2D representations.