Subgroups of $U(d)$ Induce Natural RNN and Transformer Architectures

This paper presents a direct framework for sequence models with hidden states on closed subgroups of U(d). We use a minimal axiomatic setup and derive recurrent and transformer templates from a shared skeleton in which subgroup choice acts as a drop-in replacement for state space, tangent projection, and update map. We then specialize to O(d) and evaluate orthogonal-state RNN and transformer models on Tiny Shakespeare and Penn Treebank under parameter-matched settings. We also report a general linear-mixing extension in tangent space, which applies across subgroup choices and improves finite-budget performance in the current O(d) experiments.