Non-Abelian State Tracking with Quaternionic State Space Models

Recent developments in state-space model (SSM) design have improved the state tracking and language modeling capabilities of these models through increasingly expressive transition dynamics. In particular, complex-valued diagonal transitions have been shown to be highly effective and computationally efficient. However, their expressivity remains fundamentally limited by the commutativity of diagonal matrices. In this work, we introduce Icosian, a quaternionic SSM with non-commutative state transitions through input-dependent 4D rotations. Icosian exploits the exceptional structure of the 4D rotation group SO(4), parametrized through pairs of unit quaternions. This parametrization enables fast and numerically stable recurrent computation through quaternion arithmetic, without explicitly materializing and composing rotation matrices. We evaluate Icosian on language modeling across scales from 500M to 3B parameters. Across multiple seeds, Icosian consistently outperforms Mamba-3, Gated DeltaNet, and Gated DeltaProduct, while its additional computational cost decreases with scale to only 3% at 3B parameters. We then characterize the state tracking and recall capabilities of Icosian through synthetic experiments. In contrast to complex diagonal SSMs, a single Icosian layer exhibits length generalization on the non-solvable A5 group, while deeper models also track rotations drawn uniformly from continuous groups SO(3) and SO(4). Overall, our results show that Icosian provides expressive non-commutative state dynamics and achieves notable gains in language modeling, while retaining the computational efficiency required for SSMs at scale.