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 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.

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