A projective representation is an extension of the concept of a group representation, which is commonly used in mathematics and theoretical physics. In a standard group representation, a group \( G \) acts on a vector space \( V \) through linear transformations that preserve the vector space structure. Specifically, for a group representation, there is a homomorphism from the group \( G \) into the general linear group \( GL(V) \) of the vector space.
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A projective representation is a group homomorphism into the projective linear group of a vector space. Choosing invertible linear representatives makes multiplication hold up to nonzero scalar factors. A projective unitary representation chooses unitary operators as representatives, whose scalar factors have modulus one.