representaatiotila

Representaatiotila is a Finnish term used primarily in mathematics and related disciplines to denote the space that carries a representation of a structure such as a group, algebra, or symmetry. The word combines representaatiota "representation" and tila "space" or "state".

In representation theory, a representation of a group G on a vector space V is a homomorphism ρ: G → GL(V). The space V is called the representation space, or in Finnish usage the representaatiotila, of the representation, and its dimension and structure constrain the properties of the representation.

In physics, especially quantum mechanics and the study of symmetries, Hilbert spaces (or more generally inner-product spaces) serve as representaatiotilat for symmetry groups. States of a system are vectors in the representaatiotila, while operators realize the action of the group on that space.

In computer science and cognitive science, the term is sometimes used more loosely to refer to the space of representations that a model learns or uses, such as a feature space or latent space. This usage highlights the functional role of the space rather than a single formal object.

Related concepts include representation theory, Hilbert space, group representation, and symmetry. The focus of the term is the underlying space on which a representation acts rather than the map itself.