Projective representation
In the field of mathematical representation theory , a projective representation of a group G on a vector space V over a body K is a homomorphism of G into the projective linear group :
definition
Let G be a group. A projective representation of G over a field K has the following equivalent definitions:
- It describes a homomorphism from G to the projective general linear group of a vector space over K .
- It is a map ( is the general linear group) for which there is a scalar-valued function such that
- .
Two projective representations and over a field K are called projectively equivalent if a vector space isomorphism and a function (not necessarily a homomorphism) exist such that for each and the following applies:
In other words, the two differ by a scalar multiple with a base change isomorphism.
Linear and projective representations
Each linear representation produces a projective representation by combining the representations with the quotient mapping. However, not every projective representation arises from a linear one.
Several different linear representations can lead to the same projective representation: Two linear representations are projectively equivalent if and only if one of the two can be derived by multiplying the other with a one-dimensional representation. It follows that all one-dimensional representations are projectively equivalent to one another.
Individual evidence
- ↑ B. Külshammer, M. Deiml: Representation theory . 1993, p. 54-58 .
- ^ Projective Representation. Retrieved May 30, 2016 .