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 : ${\displaystyle \Pi }$

${\displaystyle \Pi \colon G\rightarrow PGL(V)}$

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 ${\displaystyle \alpha \colon G\rightarrow GL(V)}$ ${\displaystyle GL}$ ${\displaystyle f\colon G\times G\rightarrow K^{*}}$

${\displaystyle \alpha (gh)=f(g,h)\alpha (g)\alpha (h)}$.

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: ${\displaystyle \alpha _{1}\colon G\rightarrow GL(V_{1})}$ ${\displaystyle \alpha _{2}\colon G\rightarrow GL(V_{2})}$ ${\displaystyle F\colon V_{1}\rightarrow V_{2}}$ ${\displaystyle \theta \colon G\rightarrow K^{*}}$ ${\displaystyle g\in G}$ ${\displaystyle v\in V_{1}}$

${\displaystyle F(\alpha _{1}(g)\cdot v)=\theta (g)(\alpha _{2}(g)\cdot F(v))}$

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. ${\displaystyle G\rightarrow GL(V)}$ ${\displaystyle G\rightarrow PGL(V)}$ ${\displaystyle GL(V)\rightarrow PGL(V)}$

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

1. ↑ B. Külshammer, M. Deiml: Representation theory . 1993, p. 54-58 .
2. ^ Projective Representation. Retrieved May 30, 2016 .