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

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 $U{T}_3({\text{F}}_3)$-the unitriangular matrix groups over the field with three elements. These results provide a foundation for designing new G-CNN architectures via representations of $U{T}_3({\text{F}}_3)$ that respect deep algebraic structure, with potential applications in symbolic visual learning.