Halbdarstellung

Halbdarstellung is a German term that appears in different scholarly contexts, and it does not have a single universally accepted definition. The literal meaning “half- or semi-presentation” indicates that the concept is related to a reduced or partial form of a representation, but the precise sense depends on the field and author.

In mathematics, a Halbdarstellung may refer to a partial or restricted representation, such as a representation defined only on a subspace or on a subalgebra, rather than on the entire structure. Some usages align with the idea of a projective or two-valued representation, where the representation rules hold up to a scalar factor. In this sense, Halbdarstellung can be connected to the notion of a representation that corresponds to a central extension, rather than a strict linear representation of the original object.

In physics and related areas, ideas resembling Halbdarstellung appear under projective representations of symmetry groups. These representations, which differ from ordinary representations by a phase factor, are important in areas such as quantum mechanics where spin and other quantum numbers reflect projective structures.

Because the term is not standardized, practitioners should verify the definition given in the specific text where Halbdarstellung is used. If encountered, it may indicate either a partial/restricted representation or a projective/central-extended viewpoint, depending on the context.

See also: representation theory, projective representation, partial representation, central extension.