Projective representation

In the field of <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a> in <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a projective representation of a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> G on a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> V over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> F is a <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphism</a> from G to the <a href="/facts/Projective_linear_group/y7P9rMyR">projective linear group</a>

P
          G
          L
        
        (
        V
        )
        =
        
          G
          L
        
        (
        V
        )
        
          /
        
        
          F
          
            ∗
          
        
        ,
      
    
    {\displaystyle \mathrm {PGL} (V)=\mathrm {GL} (V)/F^{*},}

where GL(V) is the <a href="/facts/General_linear_group/B54gNILS">general linear group</a> of invertible <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformations</a> of V over F, and F∗ is the <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> consisting of nonzero scalar multiples of the identity transformation (see <a href="/facts/Scalar_transformation/tjIAHrE6">Scalar transformation</a>).
In more concrete terms, a projective representation of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a collection of operators 
 
 
 
 ρ
 (
 g
 )
 ∈
 
 G
 L
 
 (
 V
 )
 ,
 
 g
 ∈
 G
 
 
 {\displaystyle \rho (g)\in \mathrm {GL} (V),\,g\in G}
 
 satisfying the homomorphism property up to a constant:

ρ
        (
        g
        )
        ρ
        (
        h
        )
        =
        c
        (
        g
        ,
        h
        )
        ρ
        (
        g
        h
        )
        ,
      
    
    {\displaystyle \rho (g)\rho (h)=c(g,h)\rho (gh),}

for some constant 
 
 
 
 c
 (
 g
 ,
 h
 )
 ∈
 F
 
 
 {\displaystyle c(g,h)\in F}
 
. Equivalently, a projective representation of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is a collection of operators 
 
 
 
 
 
 
 ρ
 ~
 
 
 
 (
 g
 )
 ⊂
 
 G
 L
 
 (
 V
 )
 ,
 g
 ∈
 G
 
 
 {\displaystyle {\tilde {\rho }}(g)\subset \mathrm {GL} (V),g\in G}
 
, such that 
 
 
 
 
 
 
 ρ
 ~
 
 
 
 (
 g
 h
 )
 =
 
 
 
 ρ
 ~
 
 
 
 (
 g
 )
 
 
 
 ρ
 ~
 
 
 
 (
 h
 )
 
 
 {\displaystyle {\tilde {\rho }}(gh)={\tilde {\rho }}(g){\tilde {\rho }}(h)}
 
. Note that, in this notation, 
 
 
 
 
 
 
 ρ
 ~
 
 
 
 (
 g
 )
 
 
 {\displaystyle {\tilde {\rho }}(g)}
 
 is a <a href="/facts/Coset/zFgQUTLY">set</a> of linear operators related by multiplication with some nonzero scalar.
If it is possible to choose a particular representative 
 
 
 
 ρ
 (
 g
 )
 ∈
 
 
 
 ρ
 ~
 
 
 
 (
 g
 )
 
 
 {\displaystyle \rho (g)\in {\tilde {\rho }}(g)}
 
 in each family of operators in such a way that the homomorphism property is satisfied exactly, rather than just up to a constant, then we say that 
 
 
 
 
 
 
 ρ
 ~
 
 
 
 
 
 {\displaystyle {\tilde {\rho }}}
 
 can be "de-projectivized", or that 
 
 
 
 
 
 
 ρ
 ~
 
 
 
 
 
 {\displaystyle {\tilde {\rho }}}
 
 can be "lifted to an ordinary representation". More concretely, we thus say that 
 
 
 
 
 
 
 ρ
 ~
 
 
 
 
 
 {\displaystyle {\tilde {\rho }}}
 
 can be de-projectivized if there are 
 
 
 
 ρ
 (
 g
 )
 ∈
 
 
 
 ρ
 ~
 
 
 
 (
 g
 )
 
 
 {\displaystyle \rho (g)\in {\tilde {\rho }}(g)}
 
 for each 
 
 
 
 g
 ∈
 G
 
 
 {\displaystyle g\in G}
 
 such that 
 
 
 
 ρ
 (
 g
 )
 ρ
 (
 h
 )
 =
 ρ
 (
 g
 h
 )
 
 
 {\displaystyle \rho (g)\rho (h)=\rho (gh)}
 
. This possibility is discussed further below.

Projective representation open-in-new

Projective representation