This term is related to: linear representation theory
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Let be a group. A projective representation of over a field is defined in the following equivalent ways:
- It is a homomorphism from to the projective general linear group for a vector space over
- It is (up to projective equivalence) a map (viz,to the general linear group) where the images of elements of are ambiguous upto scalar multiples, and such that upto a scalar multiple.
if we let be the function such that:
then we say that is a -representation.
Two projective representations and over a field are termed projectively equivalent if there exists a vector space isomorphism and a function (not necessarily a homomorphism) such that for every and :
In other words, they differ by a scalar multiplication combined with a change-of-basis isomorphism.
Linear representations give projective representations
Every linear representation gives rise to a projective representation, , simply by composing the given representation with the quotient map (which involves quotienting out by the center). However, not every projective representation arises from a linear representation.
However, it is very much possible that different linear representations descend to the same projective representation. The following is in fact true:
Two linear representations are projectively equivalent if and only if one of them can be obtained from the other via multiplication by a one-dimensional representation.
In particular, all the one-dimensional representations are projectively equivalent to each other.
Projective representation gives a 2-cocycle
Let be a projective representation. Then we can associate to it a 2-cocycle such that:
By the assumptions for a projective representation, this turns out to be a 2-cocycle from to .
It turns out that projectively equivalent projective representations give 2-cocycles that differ multiplicatively by a 2-coboundary. Thus, any projective representation up to projective equivalence defines an element of the second cohomology group for trivial group action .
When is a projective representation equivalent to a linear representation?
A projective representation is projectively equivalent to a linear representation iff the 2-cocycle associated to it is a 2-coboundary. In particular, this means that if (the second cohomology group) is trivial, any projective representation is projectively equivalent to a linear representation.
In general, any projective representation of the group gives rise to a linear representation of its Schur covering group.