Projective representation

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 $${\displaystyle \mathrm {PGL} (V)=\mathrm {GL} (V)/F^{*},}$$ where GL(V) is the general linear group of invertible linear transformations of V over F, and F^∗ is the normal subgroup 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 / F^*) via linear transformations.

In more concrete terms, a projective representation of ${\displaystyle G}$ can be represented as a collection of operators ${\displaystyle \rho (g)\in \mathrm {GL} (V),\,g\in G}$ satisfying the homomorphism property up to a constant:

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

for some constant ${\displaystyle c(g,h)\in F}$. Two such choices of operators ${\displaystyle \rho _{1},\rho _{2}}$ define the same projective representation if for any ${\displaystyle g\in G}$ the choices ${\displaystyle \rho _{1}(g),\rho _{2}(g)}$ are the same up to a scalar. Equivalently, a projective representation of ${\displaystyle G}$ is a collection of operators ${\displaystyle {\tilde {\rho }}(g)\subset \mathrm {GL} (V),g\in G}$, such that ${\displaystyle {\tilde {\rho }}(gh)={\tilde {\rho }}(g){\tilde {\rho }}(h)}$. Note that, in this notation, ${\displaystyle {\tilde {\rho }}(g)}$ is a set of linear operators related by multiplication with some nonzero scalar.

If it is possible to choose a particular representative ${\displaystyle \rho (g)\in {\tilde {\rho }}(g)}$ 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 ${\displaystyle {\tilde {\rho }}}$ can be "de-projectivized", or that ${\displaystyle {\tilde {\rho }}}$ can be "lifted to an ordinary representation". More concretely, we thus say that ${\displaystyle {\tilde {\rho }}}$ can be de-projectivized if there are ${\displaystyle \rho (g)\in {\tilde {\rho }}(g)}$ for each ${\displaystyle g\in G}$ such that ${\displaystyle \rho (g)\rho (h)=\rho (gh)}$. This possibility is discussed further below.