PROJECTIVE REPRESENTATION Meaning and
Definition
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Projective representation is a concept predominantly used in the field of mathematics, specifically in the realm of group theory and representation theory. It refers to a mathematical representation of a group that is defined up to a multiplicative factor or phase.
In a projective representation, a group is represented by operators that act on a vector space or a module. However, unlike ordinary representations, projective representations allow for the introduction of a phase factor in the transformation operators. This phase factor is typically a complex number of unit magnitude, representing a scalar factor that can affect the assigned operators. The use of these phase factors ensures that the representation retains the group structure, enabling the group's operations to be preserved.
Projective representations are especially valuable when studying certain symmetry groups, as they can capture nontrivial properties and transformations that are intrinsically connected to the structure of the group itself. They often arise in the study of physical systems, such as quantum mechanics, where they help describe symmetries that are not captured by ordinary linear representations.
Overall, projective representations provide a powerful tool for characterizing and studying the abstract algebraic properties of groups, as they go beyond traditional representations and encompass additional information derived from the group's intrinsic properties.
Etymology of PROJECTIVE REPRESENTATION
The word "projective" in "projective representation" comes from the mathematical concept of projective geometry. The term "projective" is derived from the Latin word "projectus", meaning "thrown forth" or "extended", and it signifies the extension of geometry beyond the Euclidean plane.
The term "representation", on the other hand, comes from the Latin word "representare", meaning "to present" or "to depict". It refers to the act of presenting or depicting something in a certain way.
Therefore, the etymology of "projective representation" suggests a depiction or presentation of something in an extended or expanded manner, influenced by the principles of projective geometry.
