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Concept# Projective representation

Summary

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
\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).
In more concrete terms, a projective representation of G is a collection of operators \rho(g)\in\mathrm{GL}(V),, g\in G satisfying the homomorphism property up to a constant:
:\rho(g)\rho(h) = c(g, h)\rho(gh),
for some constant c(g, h)\in F. Equivalently, a projective representation of G is a collection of operators \tilde\rho(g)\in\mathrm{PGL}(V), g\in G, such that \tilde\rho(gh)=\tilde\rho(g)\tilde\rho(h). Note that, in t

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