Representation Theory of $UT_3(\mathbb{F}_3)$ and its Applications to Equivariant Decomposition in Neural Architectures

TL;DR

This paper characterizes the decomposition of equivariant feature spaces in G-CNNs, provides explicit matrix forms for irreducible representations of $UT_3(_3)$, and discusses applications in designing algebraically structured neural architectures.

Contribution

It introduces new theorems on invariant subspace chains in G-CNNs and explicitly constructs irreducible representations of $UT_3(_3)$, enabling algebraically informed neural network design.

Findings

Theorems on equivariant feature space decomposition

Explicit matrix forms for $UT_3(_3)$ representations

Potential for new G-CNN architectures respecting algebraic structure

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 $UT_3(\F_3)$-the unitriangular matrix groups over the field with three elements. These results provide a foundation for designing new G-CNN architectures via representations of $UT_3(\F_3)$ that respect deep algebraic structure, with potential applications in symbolic visual learning.