Consider the process of collective decision-making, in which a group of individuals interactively select a preferred outcome from among a universe of alternatives. In this context, "representation" is the activity of making an individual's preferences present in the process via participation by a proxy agent---i.e. their "representative". To this end, learned models of human behavior have the potential to fill this role, with practical implications for multi-agent scenario studies and mechanism design. In this work, we investigate the possibility of training language agents to behave in the capacity of representatives of human agents, appropriately expressing the preferences of those individuals whom they stand for. First, we formalize the setting of collective decision-making---as the episodic process of interaction between a group of agents and a decision mechanism. On this basis, we then formalize the problem of digital representation---as the simulation of an agent's behavior to yield equivalent outcomes from the mechanism. Finally, we conduct an empirical case study in the setting of consensus-finding among diverse humans, and demonstrate the feasibility of fine-tuning large language models to act as digital representatives.