Linear Systems as Representations of Time Groups

TL;DR

This paper introduces a representation-theoretic framework for discrete-time linear systems, viewing them as representations of time groups acting on vector spaces, unifying various field cases.

Contribution

It develops a unified algebraic approach to linear systems using representation theory, connecting invariant decompositions and field-specific properties.

Findings

Invariant decompositions correspond to invariant subrepresentations.

Finite field systems relate to representations of finite cyclic groups.

Provides an alternative to spectral analysis in certain settings.

Abstract

In this paper, we develop a representation-theoretic formulation of discrete-time linear systems. We show that such systems are naturally viewed as representations of time groups acting on vector spaces, thereby endowing the state space with a canonical algebraic structure. This perspective provides a unified framework for linear systems over different fields, in which familiar structural properties arise from the underlying representation. In particular, invariant decompositions of the state space correspond to invariant subrepresentations, while the distinctions between real, complex, and finite-field systems emerge from the algebraic properties of the base field and the time group. We further show that linear systems over finite fields naturally correspond to representations of finite cyclic time groups, leading to module structures over polynomial quotient rings. This provides a systematic alternative to spectral analysis in settings where eigenvalue-based methods are not the most natural organizing language.