This paper aims to shed light on the evolutionary dynamics of diverse and social populations by introducing the rich expressiveness of generative models into the trait expression of social agent-based evolutionary models. Specifically, we focus on the evolution of personality traits in the context of a game-theoretic relationship as a situation in which inter-individual interests exert strong selection pressures. We construct an agent model in which linguistic descriptions of personality traits related to cooperative behavior are used as genes. The deterministic strategies extracted from Large Language Model (LLM) that make behavioral decisions based on these personality traits are used as behavioral traits. The population is evolved according to selection based on average payoff and mutation of genes by asking LLM to slightly modify the parent gene toward cooperative or selfish. Through preliminary experiments and analyses, we clarify that such a model can indeed exhibit the evolution of cooperative behavior based on the diverse and higher-order representation of personality traits. We also observed the repeated intrusion of cooperative and selfish personality traits through changes in the expression of personality traits, and found that the emerging words in the evolved gene well reflected the behavioral tendency of its personality in terms of their semantics.