Homogeneous spaces are mathematical structures where a group acts transitively on a space, meaning that for any two points in the space, there is a group element that moves one point to the other. This property allows homogeneous spaces to be studied through the lens of symmetry, making them essential in understanding the representations of groups and their actions. In the context of finite-dimensional representations and their characters, homogeneous spaces can often reveal deep insights into the nature of the representations, especially when considering how group actions preserve certain geometrical or algebraic structures.
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