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Nov 13, 2012 at 17:42 comment added Will Sawin I think the automorphism group of a nice lattice, like the $E_8$ lattice or the Leech lattice, with the standard representation, should give a counterexample.
Nov 13, 2012 at 17:04 comment added Will Sawin One potential problem with the new condition is that every finite group representation satisfies it when you add enough trivial summands. I can't immediately think of an example of a finite group representation that cannot be put in this form after adding enough trivial summands, but it seems likely that one exists.
Nov 13, 2012 at 16:22 comment added Eric Rowell Thanks Will! Including (a) was an afterthought that should have been removed upon further reflection.
Nov 13, 2012 at 15:55 history answered Will Sawin CC BY-SA 3.0