In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group
P G L ( V ) = G L ( V ) / F ∗ , {\displaystyle \mathrm {PGL} (V)=\mathrm {GL} (V)/F^{*},}
where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup of G L ( V ) {\displaystyle \mathrm {GL} (V)} consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation). Just as linear representations study the possible actions of the group G on vector spaces via linear transformation, the projective representations study the actions on lines in these vector spaces (namely (V \{0}) / F*) via linear transformations. In more concrete terms, a projective representation of G {\displaystyle G} can be represented as 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} . Two such choices of operators ρ 1 , ρ 2 {\displaystyle \rho _{1},\rho _{2}} define the same projective representation if for any g ∈ G {\displaystyle g\in G} the choices ρ 1 ( g ) , ρ 2 ( g ) {\displaystyle \rho _{1}(g),\rho _{2}(g)} are the same up to a scalar. 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 set 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.
Linear representations and projective representations One way in which a projective representation can arise is by taking a linear group representation of G on V and applying the quotient map
GL ( V , F ) → PGL ( V , F ) {\displaystyle \operatorname {GL} (V,F)\rightarrow \operatorname {PGL} (V,F)}
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