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 <[EMAIL PROTECTED]> 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 --------------------------------- Any questions? Get answers on any topic at Yahoo! Answers. Try it now. _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
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