Representationswhere

Representationswhere is a term used informally in mathematics and theoretical computer science to denote a framework for restricting representations to those that satisfy a given property. It is not a widely standardized notion in mainstream literature, but it appears in discussions about classification, constraint-based construction, and the study of definable families of representations.

Formal idea: Let G be a group (or more generally, an algebra, category, or other structure) and let Rep(G) denote the collection of representations of G. For a property P that can be evaluated on a representation, Representationswhere(G, P) (sometimes written as Rep(G) with the predicate P imposed) refers to the subcollection of representations rho for which P(rho) holds. The property P can range from structural ones, such as unitary, irreducible, or finite-dimensional, to more specific constraints like kernel containment, image in a prescribed subgroup, or dimension bounds.

Relation to other concepts: In categorical language, Representationswhere corresponds to a full subcategory of Rep(G) consisting of representations satisfying P. In algebraic geometry and representation theory, similar ideas lead to moduli spaces or stacks that parametrize representations with fixed properties, enabling the study of families and deformations under the constraint P.

Examples: Representationswhere(G, "unitary") yields unitary representations when a compatible inner product exists; Representationswhere(G, "irreducible and dimension ≤ n") selects irreducible representations of bounded dimension.

See also: Representation theory, unitary representations, irreducible representations, moduli of representations, constrained or restricted representations.

Notes: The term is informal; for rigorous work, researchers typically specify the structure, property P, and the resulting subcategory or moduli problem directly.