Representation Theory of UT₃(\mathbbF₃) and its Applications to Equivariant Decomposition in Neural Architectures

2025/07/11 by Van Nguyen, Bich, Thang, Nguyen Cao Manh
#20C35 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2507.08397

Abstract

In this paper we prove theorems characterizing the decomposition of equivariant feature spaces, filters and a structural preservation theorem for invariant subspace chains in group equivariant convolutional neural networks(G-CNN). Furthermore, we give explicit matrix forms for irreducible representations of UT₃(\F₃)-the unitriangular matrix groups over the field with three elements. These results provide a foundation for designing new G-CNN architectures via representations of UT₃(\F₃) that respect deep algebraic structure, with potential applications in symbolic visual learning.

Citations

• General E(2)-Equivariant Steerable CNNs
• Gauge Equivariant Convolutional Networks and the Icosahedral CNN
• On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups
• Learning SO(3) Equivariant Representations with Spherical CNNs
• Watch, attend and parse: An end-to-end neural network based approach to handwritten mathematical expression recognition
• Introduction to representation theory

Related

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• Intertwiners between Induced Representations (with Applications to the Theory of Equivariant Neural Networks)
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• Universal Equivariant Multilayer Perceptrons