Representation theory quick question
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Thank you! Is there any way to show that these are indeed the only subrepresentations?

Yes. The onedimensional subrep is clearly irreducible and the 2dim is as well, since rho(x) and rho(y) do not have any common eigenspace. So any other subrep must decompose as a sum of irred. reps, which must be the ones previously found. But the only options then are U, V, and U+V = F_2^3.

*any other common eigenspace

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