apply representation theory

This paper addresses the computational problem of irreducible projective representations of finite groups. Methodologically, it introduces a universal algorithm that avoids constructing the Schur covering group. Specifically, it systematically generalizes two classical algorithms to the projective setting: (i) the Burnside character algorithm—extended using projective character theory and exact integer arithmetic—to compute all irreducible projective characters and their representing matrices for a given Schur multiplier; and (ii) the Dixon numerical decomposition algorithm—adapted to floating-point arithmetic—to decompose projective representations into irreducible subspaces. The key contribution is circumventing Schur covering group construction, thereby overcoming the traditional reliance on exact knowledge of the Schur multiplier in floating-point environments. Experimental results demonstrate that this dual-path framework significantly enhances the stability, efficiency, and applicability of projective representation computation.

This work addresses the challenge of modeling permutation-equivariant representations for unaligned symmetric sets—such as unordered point clouds and graph node sets—by systematically characterizing linear layers equivariant to the symmetric group and its extension, the cyclic product group. Leveraging group representation theory, irreducible decomposition, and Schur’s lemma, we provide a unified reconstruction of DeepSets, 2-IGN, and DWS, and present the first complete classification of fully order-equivariant linear layers under the cyclic product group, revealing numerous novel non-Siamese architectures. Our framework significantly simplifies DWS derivation and removes restrictive alignment assumptions. Empirically, the proposed non-Siamese equivariant layers achieve improved performance on graph anomaly detection, neural network weight-space alignment, and Wasserstein distance learning. Code is publicly available.

This work overcomes the fundamental limitation of the Cohn–Umans group-theoretic framework—which has hitherto applied only to finite groups—by extending it to infinite groups, particularly Lie groups, thereby circumventing intrinsic barriers posed by finite groups of Lie type in matrix multiplication algorithm design. Methodologically, we generalize the triple product property and integrate Lie group representation theory with structural analysis to establish a complete theoretical framework that directly derives matrix multiplication algorithms from irreducible representations of Lie groups. Our main contributions are threefold: (1) We prove that Lie groups achieve asymptotic exponent parameters surpassing those attainable by any finite group; (2) we obtain a new upper bound on the matrix multiplication exponent ω, improving upon all prior finite-group constructions and providing a viable pathway toward ω < 2.37286; (3) we demonstrate the substantive algorithmic relevance of infinite groups in algebraic complexity theory, yielding fast matrix multiplication algorithms with concrete computational significance.

This work addresses the problem of identifying and reconstructing orbits of compact Lie group actions from point cloud data. We propose a novel method integrating geometric measure theory, computational geometry, and matrix manifold optimization to precisely recover the isomorphism type of the irreducible representation direct sum underlying an orbit—not merely detecting symmetry—and to invert the associated Lie group structure. The approach provides theoretical robustness guarantees under Hausdorff and Wasserstein distances for canonical compact Lie groups including SO(2), Tᵈ, SU(2), and SO(3). Evaluated on synthetic data up to 16 dimensions and real-world tasks in image analysis, harmonic analysis, and classical mechanics, our method achieves high-accuracy orbit identification and group structure inference. To our knowledge, this is the first end-to-end, provably falsifiable and reconstructible framework that bridges discrete point clouds to continuous group representations.

Neural operators struggle to achieve strict equivariance under non-compact Lie groups (e.g., scaling, affine groups), as their symmetries are typically characterized only by infinitesimal generators—whereas existing equivariant architectures rely on global group structure. Method: We propose Lie Algebra Canonicalization (LieLAC), the first framework to directly embed infinitesimal generators into a canonicalization pipeline, establishing a theoretical link to frame averaging over continuous non-compact groups without requiring full group representations—enabling plug-and-play equivariance enhancement. Contributions/Results: Integrated with Lie-group optimization, symmetry-driven input normalization, and physics-informed neural networks (PINNs), LieLAC significantly improves generalization and physical consistency in invariant image classification and neural PDE solving. It provides a generic, scalable solution for equivariant modeling under arbitrary Lie point symmetries.

This work addresses the problem of determining the continuous symmetries—specifically, the symmetry Lie algebra—of a parametrized algebraic variety directly from its parametric representation, without explicitly computing its vanishing ideal. We propose the first method that derives the symmetry Lie algebra directly from the parametrization, introducing a polynomial-time Monte Carlo algorithm that circumvents the high computational complexity of traditional approaches involving vanishing ideals. By integrating techniques from algebraic geometry, Lie theory, and randomized algorithms, we construct an efficient computational framework. The method is successfully applied to parametrized varieties arising in staged tree models and colored Gaussian graphical models, confirming the binomial nature of their ideals under coordinate transformations and elucidating the symmetry structures of several classes of secant varieties.

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 study addresses the problem of embedding word structures into low-dimensional complex matrix semigroups, aiming to overcome their inherent structural limitations. By integrating combinatorial word theory with representation-theoretic techniques from complex matrix groups such as SL(2, ℂ), the work proposes a novel approach that successfully constructs word representations for Euclidean Bianchi groups. This method transcends the constraints of conventional low-dimensional matrix embeddings and establishes the first symbolic word embedding framework tailored to 2×2 complex matrix semigroups. The resulting framework provides a rigorous theoretical foundation for the systematic analysis of fundamental decision problems—including membership and equivalence—within this algebraic setting.

This study addresses why language model concept embeddings exhibit specific geometric structures, such as circular or saddle-shaped configurations. Drawing upon group theory and harmonic analysis, this work reveals the intrinsic mechanism by which the statistical symmetries of data determine embedding geometry. By deriving the correspondence between irreducible representations and Fourier modes, the proposed approach generalizes translational symmetry to arbitrary finite groups and homogeneous spaces, establishing a unified theoretical framework that explains embedding geometries across diverse domains. The framework successfully reproduces known embedding structures, including the circular arrangement of months, the circle of fifths for musical chords, and spherical harmonics in astronomical contexts. These results validate the universality of the proposed theory across multimodal data spanning music, astronomy, and other fields.

This work proposes a pedagogical framework for introducing topological data analysis to students of mathematics and computer science, balancing mathematical rigor with accessibility. Departing from conventional metric-space-based approaches, the framework models data as information-carrying functions and foregrounds the role of the observer along with symmetry constraints. It naturally bridges persistent homology and symmetry-aware modeling in machine learning through group equivariant non-expansive operators (GENEOs). By integrating persistent homology, algebraic topology, and monodromy theory from two-parameter persistence, the approach forms a self-contained instructional system that significantly enhances conceptual clarity and cross-disciplinary applicability, making it well-suited for advanced undergraduate and graduate instruction.