In-context learning (ICL) is the ability of a large language model (LLM) to learn a new task from a few demonstrations presented as part of the context. Past studies have attributed a large portion of the success of ICL to the way these in-context demonstrations are represented, particularly to how labels are represented in classification tasks. On the other hand, observations of the learning capacity of ICL (i.e., the extent to which more in-context demonstrations can lead to higher performance) have been mixed, and ICL is often thought to occur only under specific conditions. The interaction between these two aspects in ICL, representation and learning, has not been studied in depth until now. We hypothesize that they are largely independent of one another, such that the representation of demonstrations determines the baseline accuracy of ICL, while learning from additional demonstrations improves only on top of this baseline. We validate this hypothesis by developing an optimization algorithm that can enumerate a spectrum of possible label sets (representations) varying in semantic relevance. We then perform ICL with varying numbers of in-context demonstrations for each of these label sets. We observed that learning happens regardless of the quality of the label set itself, although its efficiency, measured by the slope of improvement over in-context demonstrations, is conditioned on both the label set quality and the parameter count of the underlying language model. Despite the emergence of learning, the relative quality (accuracy) of the choice of a label set (representation) is largely maintained throughout learning, confirming our hypothesis and implying their orthogonality. Our work reveals a previously underexplored aspect of ICL: the independent effects of learning from demonstrations and their representations on ICL performance.