[GAP Forum] Projective representations

[Laurent Bartholdi]

A projective representation of H is really a linear representation of
the central extension H~ of H by k^* with cocycle f; this involves
lifting the representation H->GL(V) into a map H~->GL(V), and noticing
that it becomes a homomorphism thanks to the choice of cocycle.
Now V^* otimes V contains the trivial representation of H~, namely,
identifying V^* otimes V with End(V), the submodule consisting of
scalar matrices. This is not a well-defined representation of H; to
see it as a representation of H, consider an element of H; lift it to
H~; and let it act (trivially). As long as you chose the same lift for
the projective representations of H on V and on V^*, you'll get a
trivial representation of H.

On 12/3/06, D N <dn2447 at yahoo.com> wrote:
> Dear GAP Forum,
>
>   Let us work over an algebraically closed field k of characteristic 0.
> Let H be a finite group and let f : H x H --> k* be a 2-cocycle.
> Here k* is the abelian group of non-zero elements of k considered as a
> trivial module over H.
>
>   Let V be a projective representation of H with 2-cocycle f.
> Then the dual vector space V* is a projective representation of H
> with 2-cocycle 1/f.
>
>   The tensor product V \otimes V* is a representation of H.
> Is it true that the decomposition of this representation (into
> irreducible representations of H) contains atleast one copy of the
> trivial representation of H? If this is not true, then is there a way to use GAP to
>   generate counter-examples?
>
>   Any help towards answering this question is greatly appreciated.
>
>   Thanks,
> D. Naidu
>
>
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